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Some people might identify with their continent (Europeans). But this isn't convention. Middle east is technically Asian but they don't generally identify as Asian. So there's no reason for us to identify with the American continent.
That being said, I always thought that the idea of being American was something the Brits use to call the colonists of Canada and U.S.A. But after the U.S revolution, Americans kept identifying with it and Canadians dropped it to set ourselves apart (because queen, french fries, french toast, french canada idfk).