I wonder how long it will be before that stops becoming a thing, and people identify as American without the European ancestry. I don't know if it's a matter of becoming completely removed from the culture throughout many layers of generations, or that "American" needs to become a sufficient identity on the same level as Italian, German, etc.
I'm an American. My parents are American. My grandparents are American. My great-grandparents are American.
I realize that at some point in history my family was not American, but it was so long ago that it's really hard for me or any of my living relatives to care.
And even then, I identify more as an Illinoisan than an American. Sometimes I identify myself as a Viking, just because it's fun, but that has nothing to do with my relatives being from Scandanavia. I'm a Viking because my grandma grew up in Minnesota.