A hamburger is a type of sandwich made with two pieces of bread and ground-beef. Much the same way that a "Reuben", "Philly Cheese-steak", "French Dip", "Monte Cristo" and a "Hot Dog" are all sandwiches. As far as calling the meat in a burger a "burger" (as opposed to calling it a "burger patty", "ground-beef patty" or just a "patty") is incorrect. You are right in that the meat isn't a burger until meat and bread are put together.
But as far as comparing it to the way you refer to it back home, don't even try, you'll drive yourself insane. We may both speak English, but we don't speak the same language. If you ask a waitress for a "biscuit" you're not gonna get the same thing in Los Angeles that you would in London. Not to mention, things like trunks, trucks, apartments, elevators and fanny packs.
But remember: a burger is a type of sandwich.