In computer games, we have the theme that the army composition must be chosen to counter enemy units. Was that ever the case?

I would say yes and no.

Yes in that certainly some army units were employed specifically to counter enemy units. Pikes against heavy cavalry for example. Xenophon iirc mentions ad-hoc formation of slingers in the 10 000 expedition to counter Persian archers. Song dynasty infantry also employed special units armed with large swords to cut the legs of Jin cavalry.

And certainly if you only employed one type, you run the risk of easily being countered. Even if you are using them already but just don't have enough. Greeks learned the hard way they need cavalry and skirmishers. Romans learned they need good and enough cavalry.

But no in that I do not think most of the time the armies were deliberately made to be a "balanced" army. First, what even is that? Fighting is about doing what works. There are certainly situations when one would want more archers and less infantry, or more cavalry and less infantry, or more infantry and less everything else. It's about finding the best composition for the situation. In my opinion all civilizations found the best composition available for their situation pretty quickly, for if they didn't they were conquered.

So having said that, I would say, yes there were successful armies that only relied on one unit type (not saying they were exclusively one unit type). The Eurasian steppe nomads come to mind, being almost all cavalry archers. Early classical Greece also did fairly well with its reliance on hoplites (and pre-Hannibal Roman heavy infantry?) But all of these are situational.

It's also important to remember that unlike in video games, an army you recruit to fight on the steppes would most likely not be one you use to fight in the rainforest. Sorry I haven't seen a video game with that level of realism with regards to terrain effects.

Also, unlike in video games (that I have played) a lot of the times the problem is not having the resources to make a balanced army. In a video game, you can pay (say) 2000 pieces of gold and get a unit of cavalry, and this is unchanged no matter who or where you are. How the ancient Chinese wish this was true. Much of the time Chinese need to import horses and/or use steppe nomads as cavalry auxiliary. And when there's a strong nomadic confederacy that's hostile, well there goes both your auxiliary and your trade route for horses. I've read that there are times (I forgot which) where the Chinese were so lacking in cavalry horses that there was only one horse for two riders in battle.

Same with other things. The old phrase is "If you want to train an archer, start with his grandfather." You can't just pay 1500 gold every time you want a good longbow unit. If your population doesn't have the culture, tough luck. Also if you take too much casualties on your archery population, well too bad, you have less archers for the next generation.

In situations like these, I'm pretty sure having an army is still better than having no army. In other words, "yes I know having lots of cavalry to achieve a "balance" would be better, but I can't get lots of cavalry. So I will compensate by having more infantry."

As for an army of only, or mostly, knights, my opinion is that it would make sense for socio-political reasons, not military. Same with other armies composed primarily of dedicated aristocrats (samurais, spring and autumn chariots, etc). Nothing comes to mind where only archers makes sense though.