The word you are looking for is enthnicity. Enthnicity describes the (self-)perceived belonging to a population group. This is of course highly subjective.

There is undeniably perception of grouping in the US based on heritage, where it doesn’t really matter when your ancestors arrived, just from where. So from an American POV it makes sense to call him Italian, because he is in the same perceived group as all the people from Italy.

On the other hand from a European POV it doesn’t really matter, where your great grandparents come from. You are part of the US-Group, so you are American.

This is not an exclusive US Problem, but a general migration problem & it happens everywhere. Comments like yours are the reason, why people from migrated families feel like they are in-between cultures. Instead of writing snarky comments on the internet, just accept that your perception of ethnicity is part of your ethnicity and other people can have other perceptions.