When you ask for counter-play advice, and someone mentions counter-picks...

This does not mean that the only counter to a champion is playing a specific champion.

Counter-playing another champion often requires specific tools or knowledge of how they work. Frankly, it's hard enough to put into words how you are supposed to deal with certain situations, let alone to learn from just text how you are supposed to do it. There are some parts you can explain, such as timing hard CC during Katarina's ultimate, and then there are things you need to learn how to do for yourself, such as juking a skill shot, timing attacks after a signal, or avoiding specific situations.

Naming a counter-pick champion is one of the easiest ways to get all of this put into a medium that you can apply to muscle memory. Counter-picks have exactly what you need to deal with that opponent, and their kits are often either rewarded significantly for playing into the right situations or extremely adept at avoiding the wrong situations--sometimes both.

It does not mean, however, that you can't use a different champion. Counter-play is less about having all the tools that shut down your opponent, and more about using the tools you have to play around your opponent's strengths. It's entirely possible that, while practicing with a counter-pick, you might play against someone that beats you despite the counter-pick. This is because counter-play is basically about who is more experienced and made better decisions, rather than who bought what item and which champion they locked.

So, instead of complaining that you were "only" told to play another champion, start thinking about why Champ A beats Champ B.