I would say, in the US it is at the same time "not a thing" and "an even bigger thing". I guess it is easier to be seen as American, but on the other hand, those Americans claim they are Irish/German/Italian (and the corresponding percentages), because of some great-grandfather. Germany historically also has had immigrants (e.g. Ruhr area with it's mines) but those people don't think they are Polish/Romanian, because of their ancestors.