When someone from America says they Italian or whatever they aren't talking about nationality, it's about ancestry, where your family came from not what county you were born in

Dude's name is Luigi and his last name sounds like a pizza restaurant. That settles it for me, thank you very much.

According to Italian law if you have Italian ancestry, you're Italian. There's a whole process (with many asterisks and exceptions) in which you can apply to get your Italian passport

White Africans and Black Germans must have a great old time over there

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.

More of a

visits Italy for the first time over summer

continues to tell every living soul that their father's father's neighbour's goldfish, was italian

... scenario