I suppose that raises the question, if people from Berlin do indeed call themselves Berliners - do people from Hamburg call themselves Hamburgers? (genuine question)
I don't see why not. Hamburger means "of Hamburg". If you're thinking about the food item, it isn't a funny coincidence or anything, its name is derived from Hamburg.
Compare "Danish" as an American term for a Viennese pastry, "he's Danish" doesn't sound odd.
We don't generally abbreviate foods named after places, but:
"Cheddar is 200 miles from both Sandwich and Stilton."
These are all places in England.