Yes. 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:

Yes. 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.