The United States of America is the only country in the world with "America" in its name, hence we're called "Americans". (Likewise, Great Britain is actually the "United Kingdom of Great Britain and Northern Ireland", which means if anything, calling UKians "British" actually excludes part of their population.) If you follow the general rule for naming nationalities (take the country's official name, drop any preceding "People's/Democratic/Federal/Socialist/United Kingdom/States/Republic/Empire/Emirate of", and add "ish" or "an" or "ian" or "n" or "ese" to it or drop the last vowel depending on what sounds best) and apply it to "United States of America", you get "American". Sorry we picked a lame and generic name. I think we considered "Columbia" at one point but it's too late to change it now and people would confuse us with Colombia.
"USians" is technically more ambiguous since we're not the only "United States"--Mexico is "Estados Unidos Mexicanos", which is Spanish for "United Mexican States".
As you may or may not have noticed, the names of countries don't always exactly match every single geographical use of the same word. Some people in Greece were similarly upset about Macedonia, which is why lots of people call it the "former Yugoslav Republic of Macedonia". I don't think anyone ever calls their citizens "Former Yugoslav Republicans" rather than "Macedonians", but my Greek isn't up to finding out for sure.
If you want to go further back, "Germany" and "Spain" are Anglicizations of "Germania" and "Hispania", which in the original Latin incorporated much larger areas than these current countries. I don't think many Polish or Portuguese people really mind, though, when I don't refer to them as German or Spanish despite the fact that they live in Germania or Hispania.
 Historical reason: of the original 13 colonies, they each had their own names, and the only thing they really had in common were that they were United in opposition to the Crown, that they considered themselves States, and that they were of America. By the time they had become a single nationality, it was too late to pick a more interesting name.