The funny part is that you don't have a name that only applies to people from your country. As the other comment said, there many millions of Americans that are not from the US.
In the same vein, people from the EU call themselves Europeans even though there are many millions of Europeans that don't live in EU countries.
To answer your question: I identify myself as European, Spaniard and Catalan. Just like I'm pretty sure there will be some people in the US that consider themselves Americans and Texans or Californians or New Yorkers or whatever...
Technically, Mexicans and Canadians (and others) also live on the continent of North America. Lots of other not-USA-people live on the continent of South America.
In the same vein, people from the EU call themselves Europeans even though there are many millions of Europeans that don't live in EU countries.
To answer your question: I identify myself as European, Spaniard and Catalan. Just like I'm pretty sure there will be some people in the US that consider themselves Americans and Texans or Californians or New Yorkers or whatever...