I think the fact that we are a relatively very youg country has a lot to do with it also. Most Americas only have to go back 4 or 5 generations to trace their roots back to a European country.
I agree with their being a difference in claiming your heritage and claiming to be actually from a different country. It is also silly to pretend it is only the white Americans that do this. The term European American isn't heard nearly as often as African American or Mexican American.