If someone was born and raised in Poland, for example, then went to the United States and lived there, they would be considered Polish American.
My story, however, is a bit more complicated. I was born in Chile and stayed there with my family until I moved out as a child. Then, I lived in Brazil until I was 13 and moved to the United States. When I was 17, my family and I moved to France. Am I just Chilean-French or something? Chilean?
If asked, I would just say « Je suis né au Chili mais je réside à France maintenant. »
It depends on how you feel. For instance, I have a few friends that are born in Norway, but their parents are Vietnamese. Those friends would always described themselves as Vietnamese while in Norway; they are a part of that community, their ancestry is important to them etc. While traveling abroad however, they would say they are Norwegian. The context is different, and citizenship is more relevant. Some of them have lived several places, but their roots and citizenship were the most important pieces of nationalidentity to them.
Myself I’m an immigrant. I don’t tell people I don’t really know what county I came from; it’s private and not relevant. If people ask me (not abroad), they usually mean my accent is off, which is rude.