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.
I think what you mean to say is "fast food" comes from America.
"In the mid-1700s, “The Art of Cookery Made Plain and Easy” by Hannah Glasse carried a “Hamburgh sausages” recipe, which was served on toasted bread. In Germany, a meat patty on bread called Rundstück Warm was popular by at least 1869."
A rundstück warm I got at a bazaar in Germany was one of my favorite things I've ever eaten, but I wouldn't call it a type of burger, nor the meat inside a patty
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.