People from the United States of America are called (in English) "Americans" for about the same reason that people from Estados Unidos Mexicanos are called "Mexicans"--that is the most suitable general designation for those people that is distinctive from how people from other countries are designated. The usage in Spanish is a little bit more complicated, but there isn't much confusion about how to write English on this point.

The use of "American" as a stand-alone term, for better or for worse, means a person from the United States. It persists largely because of the difficulty in producing an ajectival (demonym) from the term United States. Questioning it now, when there is clearly no real confusion (like, for example, Indian/American Indian/Native American/Indian American) is prescriptivist and hair-splitting.

Also, American and Western culture are used practically interchangeably. I find this quite ironic.