The term you are looking for is 'Demonym' or sometime the word used is gentilic.
I like demonymn myself because that was always the lable givin it in the CIA World Factbook. It's what we call people based on their national citizenship (sometimes applied to a person based on their "home-country").
So why do Citizens of the United States get to be called 'Americans'?
Mostly it's a matter of convenience. United Statesians wouldn't work because there are other 'United States' out there, Mexico is actually named "United States of Mexico". So a demonym is something everyone more or less agrees on, it should specify the nations geographic location, and it has to be unique. America is a part of our nation's name and it fixes us to a specific if broad location. Add to that, Mexico's citizens have been happily calling themselves Mexicans for a good long time and Canadians are just as happy being called Canadians.
Put it this way, it's one of the few times American arrogance didn't just claim a continent for itself (or at least a continental name).
Some fun with demonyms:
- People from the Netherlands are most often called Dutch
- People from Guernsey are called Sarnian
- People from The Hague are called Hagenaar or Hagenees depending on being born there or not.