You know, I've heard this, as far as I know, recent complaint that we refer to ourselves as Americans, when supposedly everyone who lives in this hemisphere is an American. It's actually a false argument and faulty reasoning.
From the Declaration of Independence forward, the full name of our nation has been the United States of America. Even well before the Declaration, the colonists were referred to by both themselves and the world powers of the day as "Americans." Given the full name of our nation, what else would we becalled: Unitarians? Statesians? What? The only name formulation that fits easily is Americans -- that is, citizens of the United States of America.
Virtually all of the other countries in this hemisphere have one or two-word names that make it easy to refer to their citizens in a way that smoothly identifies their nationality: Cubans, Canadians, Mexicans, Columbians, Puerto Ricans, Bolivians, Argentinians, Hatians, and so on.
However, our unique history, in which we went from 13 separate colonies to 13 sovereign states, to federated states and territories which later became states, makes the umbrella name for our union of states awkward in terms of formulating a short-hand name for its citizens. Only "American" fits, and only "American" has been in continuous use since colonial times.
I think the folks in Latin America refer to the people of Canada as "canadians".
Thanks for the historical perspective. I hadn't thought of what the alternative might be if we didn't go by the title "Americans."