Speaking for the Ancient Greeks here - they did not call them anything.

The world of Greek warfare was a world of militias. There were no professional armies, military academies or officer cadres. Greek armies were brought into being for the sake of a particular mission: invade neighbour X, assault town Y, confront army Z. Once that mission was completed, the army returned home and was disbanded. At that point it simply ceased to exist. No structural element of the army that had fought a particular campaign was preserved for the next one. The men returned to their homes, the officers lost their ranks, any special equipment was dumped in the state arsenal or wharf, and that was it. There would barely even be a paper trail to prove that particular citizens had fought under particular officers at particular times.

When another campaign needed to be fought, a totally new army would be created. It would be recruited from the same pool of citizens and resident foreigners, of course; it would be organised into the same number of subunits, since these were usually tied to administrative districts to make sure the burden of military service was equally distributed. It might even be commanded by the same officers and generals, if they were re-elected to their positions. But it was not the same army. It would not have the same spirit or the same symbols or the same traditions, except in the sense that people from the same community would bring those things with them from their regular life.

In this military environment, numbered or named units would make no sense. Why number or name your units, when they only existed for a short time and the only thing that set them apart was the purpose for which they had been raised? Very few states could ever afford to field more than one army at a time, so there was no need for labels to tell them apart. Militia armies received no training and were not capable of complex operations, so there was little need for the rationalisation of divisions within a column; if needed, it could be arranged ad hoc.

Even the subunits created for the purpose of quick and equitable recruitment from districts did not really recieve names or numbers. At Athens, the hoplite body was divided into ten taxeis of just over 1000 men, one taxis for each administrative tribe. They were not numbered or named, and simply referred to as "the men from the tribe Aiantis" and so on. For each campaign the men serving for their tribe would be different. At Sparta, at least in the 4th century BC, the army was divided into six morai, each about the size of a modern battalion. There are occasional references to "the first mora", but this doesn't seem to be an official designation attached to a certain mora but simply indicate whichever mora was with the king at the head of the marching column. Our sources on these units are so bad that we can't even tell if the Spartan royal bodyguard, the famous 300 hippeis, were part of the first mora or were drawn up separately.

There are a few rare examples of armies that had a longer service record than a single campaign. These would normally be Athenian expeditionary forces, garrisons, or mercenary bands. In no case do they develop a formal numbered designation. Mostly they will be referred to by their task ("the men at Potidaia"), their location ("the garrison on the Kadmeia") or the name of their commander ("the men of Cyrus", "the men of Brasidas").

Besides these, the only units to receive special designations were a very small selection of elite standing units. I've already mentioned the Spartan hippeis ("horsemen"), 300 picked men who formed a royal guard. The other famous example is the Theban hieros lochos (Sacred Band), though the name only appears in late sources; Xenophon refers to them simply as "the picked troops of the Thebans". There are a couple other examples, such as the Thousand (an Argive standing unit), the Skiritai (an Arkadian levy in regular Spartan service) and the Arkadian eparitoi, a short-lived professional army whose name is obscure. None of these names reflect any naming system or convention; each is unique. Many other picked units were referred to generically as lochagoi or epilektoi ("the chosen ones").