See its really easy, The United States of AMERICA. The country is America and it is made up of a group of United States. Thus American's. If you have an issue with that please dig up Amerigo Vespucci. Canada is is not Canada of America, Mexico is not Mexico of America. Since the USA was intended to operate in a more states right manner. Like an EU but with a bit more power to the central government is makes total logical sense that people living in the states that make up the union are American's Kind of like
The United Kingdom is a country that consists of Great Britain and Northern Ireland. In fact, the official name of the country is "United Kingdom of Great Britain and Northern Ireland."
So they are called British not Ukians. See how that works.