Minced, spiced beef cooked as a patty was called “Hamburg steak” or “Steak in the style of Hamburg.” Easy step to calling the same meat on a bun a “hamburger.” As for “frankfurter,” per Wiki: The word frankfurter comes from Frankfurt, Germany, where pork sausages served in a bun similar to hot dogs originated. These sausages, Frankfurter Würstchen, were known since the 13th century and given to the people on the event of imperial coronations, starting with the coronation of Maximilian II, Holy Roman Emperor as King.
What is referred to generically as a hamburger–a sandwich of cooked ground beef in a bun–does seem to be an American invention. Now, whether you want to call it European because the original patty might have been derived from Hamburg steak or frikadelle or whatever, that’s your prerogative. I consider the sandwich form an American invention, and it is very much culturally identified around the world as American.
I am wondering now who thought up the idea? Humans saw animals eating other animals, they may have even tried some themself. It probably didn’t taste good raw. Who thought to cook it and make it palatable?
Actually, they were around on menus before that–I think I’ve seen sites as early as 1880s in Tulsa, OK. samclem might be better able to say, as he mentioned in the linked-to thread that the 1904 Worlds Fair was certainly not the first time a Hamburger sandwich was served in the States.
He didn’t make a hamburger sandwich, though. The American “hamburger” is an amalgamation of the German Hamburg steak (and you can go farther back than that, if you want, to the Tartars) and the Earl of Sandwich’s sandwich. Is that better?