FWIW, I have never encountered the concept of the "American race". Are you sure it's actually a thing? To me the line from American Dad's theme song seems like a joke (since America has many races).
AMER'ICAN, noun A native of America; originally applied to the aboriginals, or copper-colored races, found here by the Europeans; but now applied to the descendants of Europeans born in America.
AMER'ICAN, noun A native of America; originally applied to the aboriginals, or copper-colored races, found here by the Europeans; but now applied to the descendants of Europeans born in America.