In school, Americans study English. It's not called American English. There are some small differences in spelling and vocabulary (and of course pronunciation) when we compare it with British English, but in English classes students don't study these differences. They study grammar, writing, literature, etc. and that will be almost exactly the same in US, UK, Canada, Australia, and all of the other countries with English as the language of instruction. A lot of the differences are not things that we learn in class; they are just variations of English. For example, in American schools, we don't learn that Americans say apartment and the British say flat. We know the word apartment because we've grown up in this country. Usually Americans don't know this meaning of 'flat' unless it's through movies, reading, friends, etc.
Spelling is taught formally, but again, it's not taught as "American spelling." It's just spelling that happens to be the American spelling because the kids are in the US, and that's the variation here.
So really, most of the differences that you notice between British and American English aren't differences that have to do with the schools. They are just variations of the same language that students learn just because they grow up in a particular place in the world.