Citizens of the USA customarily refer to themselves as "Americans", McGrath....perhaps because it takes too long to say: "United Statesians" ;-)

"The United States of America" is an odd name for a country, but it derives from the fact that there were 13 colonies who joined together to fight the British, and they all considered themselves to be independent sovereign states, but joined in a federation.

Similarly, Holland called itself "The United Provinces" when several Dutch provinces joined together to throw out their colonial rulers from Spain. If they'd called themselves "The United Provinces of Europe", it would have been about as odd as what the American states decided to call themselves. America does not end at the USA's borders, it comprises everything from the North Pole to the tip of Tierre Del Fuego.

The whole name of Mexico, by the way, is "The United States of Mexico" (Los Estados Unidos de Mexico). That works a lot better, because it makes geographical sense.