It helps thinking about it the other way around, ie. nationality has DNA. For Americans this may sound absurd, but it's very common in non-immigration heavy countries. Whether "German" or "Norwegian" are nationalities or ethnicities is not a settled debate. This is why being non-white in Europe can be hard in some ways. Some Germans have a rather uncharming word for people with their nationality but not their ethnicity: passport-German. The meaning being that you're not really German, you just have citizenship, which is far less meaningful. I'm not hating on the Germans, btw; most countries are worse than Germany. I chose it as an example because they debate a lot about this, so everything's really visible. Just know that the US is the exception.
I was actually shocked by the levels of overt racism I saw when I lived in Berlin a few years back. I was living with a family as part of a foreign language exchange program, and the mother was a elementary school teacher. She would baldly say things about Turkish immigrants or Africans that would take me aback. I'm not from the most progressive and enlightened part of the country, but even my racist and crotchety grandfather wouldn't be quite that open about it.
Makes me wonder whether Americans really have that big of a problem with racism (relative to other countries), or whether they just talk more about it than others.
Btw, if you're interested in what it's like being black in Germany you should check out http://schwarzrotgold.tv/en/ . It's a series of interviews with black Germans, with English subtitles. The front page video is an interview with a mixed guy who as a child worked in an ethnic circus/display of a sort and in his youth was actually kicked out of the Hitlerjugend for being black. He's really thought a lot about this. Extremely fascinating. Tried submitting to HN, but no traction.
Ok, but what makes an American an "American" besides nationality? I'm "American" if my ascendents have been living and breeding in the US for N generations? Are the Mexican immigrants that have been living in US soil for generations also American? Or some people just have DNA mutations that happened in America?
This is non-sense from every angle you look at it and a petty excuse for racism.