Why? It would be so boring in there, I mean what does he do in that cave all day? Maybe there is some sort of amazing treasure in that cave that he's just trying to hide from us. But all joking aside, why does he choose to stay in that cave for the rest of his life?
I'm not going to begin to question the thought process of a completely non-human entity, let alone one that was genetically-engineered to be a heartless super soldier, but if I had to guess, I would say that it's because Mewtwo was created to fight, and Cerulean Cave is where you can find the strongest Pokémon in Kanto. It's similar to how, in
Kyurem vs. The Swords Of Justice, Keldeo's recognition as a Sword is dependent upon his ability to combat Kyurem, who is said to be the strongest Pokémon in Unova (and, incidentally, also secludes itself within a cave).
Also it makes you question the parents. Instead of sending their kids to school they let them go on an adventure around the world. Then again maybe they just don't like us. That would be the easiest way to avoid a college fund.
"Wait what you want to be a doctor. NOPE your going on an adventure around the world."
Completely different universe, completely different morals. And like I said, it's usually the kids who want to go off on the adventure. The parents
do not force them into it. As for school, it can be assumed that the player does still attend school to some capacity, or is at least home-schooled. Otherwise, there wouldn't be Trainer schools featured in-game.