I really only hear people discuss heritage when a persons' appearance or last name makes it ambiguous and they're asking out of sheer curiosity (this happens to me often). After all, there is no "American" look as far as physical features go.
I always find it curious that when the US has some unique, cultural "phenomenon" it is usually viewed with a critical eye from other countries. If Americans had the same attitude towards them though, they'd be called narrow-minded, ignorant, etc. The double standard seems to make it okay to be bigoted towards Americans. I find that ridiculous & hard to understand.
I mean, so what if Americans have an unusual curiosity & interest over people's heritages? So what if they recognize subtle nuances amongst one another culturally that they link to their heritages? That's all anyone means when they say "I'm Italian" or whatever - they've noted patters in similarities/differences related to heritage. Since a lot of communities are initially based on people of the same heritage congregating in one area, it makes sense that you have micro-cultures (even more so when you consider the physical size of the country). Of course, these begin to be absorbed into general American culture over time, but the more recent the immigration, the more it has a distinction from the generic culture. Still, some nuances can remain. I think the tendency to be curious about it is just indicative of being curious in general, which I consider a positive thing.
The high concentrations on this map of certain ethnic heritages in some areas is no coincidence in relation to the "micro-cultures" associated with the region: