Burgers were invented in America.
Now sure, the Hamburger is named after Hamburg in Germany. However, the German version is actually the Hamburg steak. In other words, the meat patty. And a patty does not a burger make.
America was the country that put the bun on the hamburger. This, in drag’s opinion, is when the Hamburg steak became the “burger”. A new word, losing the ham and becoming what would be translated from German as “The people’s meal”. An apt description for the American burger.
In all my time I spent in Hamburg I have never seen anyone claiming they’d serve a genuine Hamburger or something along those lines. Neither have I seen anyone else in Germany do so.
The closest German food to a hamburger would be a “Bulette im Brötchen”. Which is much closer to a big meatball than a typical burger patty. And nobody calls them “Hamburger”.
So I’m actually more inclined to believe that the hamburger got its name from Hamburg in the USA.
Edit: Similar to “German Chocolate Cake” which is also not found anywhere in Germany because it was invented by an American named German.