You're right to be skeptical of this common way to model premodern warfare. Probably the best way to look at the tactical interplay of different unit types in Greek warfare (which I'll focus on) is that the "rock-paper-scissors" model is somewhat true but also largely irrelevant.

I'll start with the somewhat true. The Classical Greeks were very aware of the doctrine we now call combined arms warfare: the notion that different troop types have distinct strengths and weaknesses, and are therefore most effectively used in combination. It's not really as simple as rock-paper-scissors, but the principle is there. For example Greek hoplites are effective in close combat, but they are slow and vulnerable in open ground. Greek archers carried no armour and small shields at best, so any well-equipped warrior who can get close to them will be at a huge advantage. Therefore light infantry is most effective when it can withdraw on a hoplite formation; hoplites are most effective when they are supported by light troops that can keep ranged units at bay. The apex predator of Greek warfare was the horseman, whose superior mobility and versatile weapons allowed him to ride down almost any opponent while easily avoiding a fight he might not win.

But the response to this understanding of strengths and weaknesses was usually not to develop "counters" or to "invest" in a particular weapon type. Instead, what states and commanders would do in order to address tactical shortcomings was to strive to deploy anything they could that might help. There was no single cure-all counter to all situations; there was no weapon system that could be relied upon to always defeat a particular opponent; and even if there were, it wasn't always available and couldn't always be bought. The only solution to the constantly changing conditions of actual campaigning was to field a flexible army that could acquit itself well in any crisis. It was not necessarily the weapons of the enemy army that made you most vulnerable, but the weapons you lacked in your own.

So, for instance, the Athenians who sailed to invade Sicily in 415 BC knew that they would be up against the renowned Syracusan cavalry - a force that could tactically and strategically paralyse them if they could find no way to neutralise it. But they did not do so through the computer game solution of fielding more spearmen. In fact they had hoplites aplenty - that was exactly the problem, because hoplites were too slow and vulnerable to outflanking. Instead, the Athenian generals set out to hire Thracian peltasts and Cretan archers and Rhodian slingers and to bring horsemen of their own (Thuc. 6.43). In short, they turned a vulnerable single-strength army into a combined-arms force that might stand a chance (though in the end, the Syracusan horsemen still proved utterly superior).

Another example. When the mercenary army called the Ten Thousand found itself trapped in Persian territory in 401 BC, it soon realised that its contingent of 2500 Thracian peltasts and a few hundred Cretan archers simply didn't have the range to deal with the Persian archers and slingers. The computer game solution might be to get as many long-range missile troops as possible, or maybe some artillery (but unfortunately that was still 2 years from being invented). Indeed the mercenary army rearmed some Rhodian hoplites as slingers, who found that they could outrange their Persian assailants and keep the enemy at a distance. But they also tried to find ways for their Cretan archers to shoot further and scraped together a force of 40 horsemen to lead charges against the enemy light troops (Xen. An. 3.3.15-19). Again, they didn't make a beeline for a hard counter, but did everything they could all at the same time.

There are several reasons for this approach. One is the simple pressures of time and money. Even if a state or general knew the perfect "counter" to the enemy's preferred weapons, there might not be time to train the troops or money to get the equipment. Greek states had no standing armies, so there was little any leader could to to prepare in advance for a coming war; most armies were only raised when the campaign was declared, and could not be financially supported beyond its duration. Commanders had to work with the tools they were given. Only especially rich states like Athens could use public funding to maintain professional forces to make up for the shortcomings of their militia.

A second crucial factor is custom. Greek militias had no regular training of any kind; their military strength relied on an inert civic sense of "how we fight" which made their levies moderately effective at one particular way of war but deeply inflexible and unwilling to change. People aren't easily made to fight in a certain way just on the basis of vague promises that it'll work better. Indeed, Greek hoplites had an innate cultural resistance to training as such, which they saw as ridiculous and unnecessary; it's only when the Spartans and (more importantly) the Macedonians showed them the merits of a highly trained force that they began to adopt it for themselves. It was usually easier to hire specialists from elsewhere than to retool your entire established way of war. For instance, the remains of the Ten Thousand mentioned above were eventually hired by the Thracian king Seuthes, who realised that he had plenty of horsemen and light infantry, but lacked the sturdy backbone that hoplites could provide.

Related to this is what you might call the extent of the tactical imagination. As I said earlier, the Greeks were perfectly aware of the strengths and weaknesses of different troop types and the ways they could be used to mutual benefit. But this only covered the troops they were familiar with. There were no military academies or professional officers, so to some extent every commander had to reinvent the wheel; they were very unlikely to come up with anything that was radically outside the established forms of war. In other words, while Greek commanders were well able to find ways to use peltasts and slingers to advantage, they would not have come up with the concept of the legionary. Tactics are an aspect of culture, and tactical thinking is constrained by its presumptions and not completely free. Commanders are most likely to look for solutions to tactical problems in the examples they know from earlier battles. Real change is usually slow or driven by exceptional circumstances.

But probably the most significant point (at least from the point of view of tactical practice) is that the Greeks understood that there were many kinds of military engagement, and many surprises and unwanted developments in war. You simply couldn't predict what kind of battle you might end up in, and you'd rarely have the luxury of only encountering one type. As a result, it was far more important for armies to be flexible than to be very effective at resolving one particular tactical problem. Different warrior types didn't just have generic strengths and weaknesses; their strengths and weaknesses varied by context. Xenophon explains that the marching column should be organized differently depending on what terrain it's passing through (rugged or flat, wooded or open), on the weather, whether it's day or night, and so on. It was impossible to train any single warrior to excel in all these environments. The best possible solution was to field an army with all imaginable parts and use them as and when they were most effective.

 

(The idea that certain troop types and fighting styles were considered unworthy or dishonourable is less relevant, as it doesn't seem to have affected the Greek way of war very much. I wrote more about their affected disdain for missile troops in this older answer.)

Some of these ideas are intuitive and we do see a historical basis for them. Trying to use spears to fight against cavalry is pretty obvious and there's evidence that generals did think about countering tactics of their opponents. Note that I said countering tactics, rather than units. This is important. Units, as they are modernly depicted, are too neat and organized. The historical reality was much more complex, having a neat unit of combatants on hand to counter the opponents neat unit of only melee cavalry didn't happen that often. Battles were complicated affairs, and generals had to make do with what they had on hand to counter a wide variety of different situations that might materialize mid battle.

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The idea that an ancient or medieval army is broken up into discrete units of homogeneous combatants has some light historical basis but is largely a modern construct. While the Roman maniple system is an outlier, generally armies were organized in a line of battle. The line was organized in the camp, with the first combatants marching out of the camp making up the far flank, the next ones the center, and the last, the near flank. There might be several lines of battle, front and reserve line being common, but there wasn't a sense of having a large number of small units that could be commanded independently. These battle lines form units in the modern sense, but an entire army could just be a single battle line. Pre-Marian Roman armies fought in the acies triplex formation which is three of these battle lines. There wasn't a great deal of control a general could apply to these battle lines, once the first line was committed, that was it. Many generals would stay with the reserves (e.g. Caesar, Alexander) and attempt to influence the battle by their application. However, Gaius Marius took up a shield and stood in the center of the first line against the Cimbri and Teutones, knowing that morale was going to be more important than any kind of complicated tactic he could orchestrate from the rear. In summary, armies were divided into a few very large units, but they were not homogenous and couldn't be controlled or maneuvered easily.

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Regarding equipment, a general would muster the soldiers he could, and take what they were equipped with. Soldiers from different cultures or geographies were generally equipped in a certain way and used to fighting in a certain style. An early Imperial Roman army might be made up of relatively homogenous legionaries from the core provinces but might also be supported with auxiliaries. The idea to combine Germanic auxiliaries equipped with spears into units with Numidian auxiliaries also equipped with spears would not have occurred to a general of the time. The Germans and Numidians would fight together with their comrades instead of trying to create these arbitrary units. If the German auxiliaries turn up with shields, spears, slings, and javelins where would they be categorized anyway? How the combatants were used to fighting, whether it be skirmishing, or close combat, etc. was more important than what every individual was equipped with. In the later Byzantine military manuals, soldiers were kept together with close kin and family. When they joined the battle line, they would be surrounded by people they knew, rather than people equipped in an identical fashion. Obviously if one person in the family unit can afford a horse, they might be given a different role but that seems to be more of an exception than the rule.

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This issue only gets more complicated as we move forward in history into the middle ages. The more sophisticated ancient states could centrally fund a standing army of a relatively homogenous nature. Alexander could rely on a standing force of phalangites to act as the core to his force. Although they might be supplemented by other units in different situations, he could at least rely on a consistent line of battle for every engagement. In the late antique and early medieval period, Roman generals (with whom I am most familiar) had to piece together incredible disparate and heterogenous armies. Even imperial troops could not be relied on to have uniform equipment. In Maurice's Strategikon, he suggested that when forming the line of battle, soldiers who were equipped with shields and armor were to stand in the front ranks of the line. Ideally the first 2-3 ranks would be armored. The following ranks might have to be filled with unarmored combatants depending on the soldiers available. Maurice also describes common opponents of the time, the Persians, Avars, Franks, Lombards, Slavs, but never describes specific units. Persians fight in an ordered line, are good at shooting weak bows quickly, can bear heat well but are hurt by cold weather, they are not able to attack and retreat quickly. The Franks, Lombards and other "fair haired" people are brave and strong in direct engagements, but vulnerable to skirmishes, ambushes and surprise attacks. Again, these are more general statements, rather than how a modern game might lay out a faction as having strong archers, so cavalry are needed to counter them. A general probably would not have thought about putting a unit of spearmen on a flank to protect against cavalry. Instead, a general might make sure his infantry had spears, in the event they were attacked by cavalry. For example, at the Battle of Pharsalus, Caesar knew his cavalry were hopelessly outnumbered by Pompey's and instructed his infantry not to throw their Pila (javelins) but to use them as spears instead. This seemed to be effective as the cavalry broke and fled after an engagement with Caesar's infantry.

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One of the only specific unit types I can recall being mentioned with a needed counter are the heavy cavalry of the Arab emirates and Caliphate in the 9th/10th centuries CE. There are several Byzantine military manuals that lay out very specific instructions for a specific type of soldier, the menavlion to be kept in reserve waiting for a frontal heavy cavalry charge. They are to be strong and brave, and equipped with a menavlion. This weapon was cut from a single sapling, made as large as hands can hold, with a long blade (18 inches) on the top. The blade needed to be reinforced with strips of iron to firmly attach is to the shaft. When the heavy cavalry were approaching, the menavlions would plant to rear of their weapons into the ground and form the first rank of the infantry formation. Their job was to impale the horses during the charge to break up the attack, while their comrades behind went after the riders. That being said, the middle Byzantine army was not organized anything like modern representations. The entire army was formed into a square, with kataphracts, the general, javeliners and the baggage train inside, and each side of the square being made up of 7 ranks. 2 of spearmen, 3 of archers, and 2 of spearmen. The entire army moved in the square at almost all times. javeliners and reserve units would reinforce different areas during the battle but the square would remain as constant as possible, even when marching. The key Byzantine tactic of the time also breaks the neat rock/paper/scissors aspect of modern representations. The kataphracts were to be organized in a wedge and charge directly into the center of the enemy's infantry who were generally spearmen. The armor of the kataphracts and their horses was able to shatter spears without harm to their bearers and this tactic won many victories in the 10th and early 11th centuries. Before you say that the kataphracts here are an example of a homogenous unit, prokoursatores or coursers (skirmishing horsemen) shielded the wedge from the front and flanks on the approach, and also made up the rear of the formation.

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In conclusion, there are examples of neat units in ancient and medieval armies, but they are not a common feature across time periods and cultures. Where units do appear, they are organized on a tactical or cultural level, rather than on equipment. There are also examples of certain tactics or equipment being useful in specific tactical situations, but it is more nuanced than rock beats scissors. For every Caesar beating off Pompey's cavalry with pila used as spears, there is another example of kataphracts charging straight into spearmen and winning, or Roman legionnaires charging straight into a Macedonian phalanx and coming out victorious (Battles of Pydna and Cynoscephalae).

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Strategikon Maurice Tiberius

Praecepta militaria Nikephoros II Phokas

Taktika Nikephoros Ouranos

History of Rome Livy

Life of Alexander and Life of Caesar Plutarch

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edits for grammar

I asked this question once and, in addition to the great answers already here, you might want to look at u/wotan_weevil's response to my question: https://www.reddit.com/r/AskHistorians/comments/b81zd1/comment/ejvgz2t

The "counter" mentality is really a game mechanic. Actual military operations were more influenced by organisation and logistics, which in turn dictated the battlefield tactics. If there was a cookie-cutter template, it would be the deployment of your infantry in the centre, skirmishers at the front and light troops/cavalry at the flanks to prevent the army from being surrounded. Exactly what weapon was used was not the focus of a commander.

Games like the Total War series plays up the notion that each unit had one type of weapon - for example, you have "katana samurai" who can defeat "yari samurai". But in a modern context, that would be like having "pistol infantry" countering "sniper infantry" who can defeat "assault rifle infantry". The reality was that soldiers carried multiple arms and used them as necessary. The spear and other polearms were the general battlefield weapon - cost efficient, easy to use for most fighters and the most versatile when fighting in formation. Swords were more commonly carried in a way we regard as a sidearm.

Thus, a commander wouldn't go "Hm, those German barbarians have SPEAR infantry. I will send my SWORD infantry to counter them".

The unpredictable nature of battle would mean that any "counter" could simply fail given the right command, morale, tactics, training and a bit of luck. Logically, spears counter cavalry - but only if the formation holds firm. If the formation loses cohesion, cavalry breaks the line.

The English mindset was mostly around fighting dismounted with their men-at-arms in the centre. The counter to French cavalry? Archers. A few thousand shafts into a charging line will bring down horses and men. It worked at Agincourt when the archers held the line. It didn't work at Formingny when they left their positions and were counter-attacked.

Agincourt also has the exceptional mention of archers joining the melee with mallets and swords, effectively fighting as light infantry against the French infantry.

In summary, specific counters were probably not in the forefront of commanders unless they had reason to know of specific tactics used by the enemy from prior experience. This was mostly around adapting their tactics rather than equipping their armies in specific fashion.