Allegedly the idea was brought to the US by immigrants from Hamburg, thus called Hamburger Steak, but I have no clue how accurate it is. They certainly are made everywhere though, not just (or at all? Not sure if those 12 cows that are still left in Hamburg provide enough meat for local production) in Hamburg.

I actually think that the Fischbrötchen (bun with fish) that is a thing in Hamburg might have been the origin but it doesn't explain how the ground meat patty moved in really and giving the whole thing the name "hamburger" must have happened in America.