Entering this discussion very late - and I'm Canadian, not American - but I'm guessing that some Americans get very precious about the "I'm Irish" sort of thing and that would annoy a lot of Europeans. ESPECIALLY if the person who is identifying themselves as Irish, or whatever, has a real lack of knowledge about the country they claim to identify with - I can easily imagine that happening.
Irish is probably a particularly fraught area. Obviously there are a lot of Irish-Americans but for quite a long time (and probably still somewhat) if you were really keen on identifying yourself as Irish in the US it may have had something to do with the IRA or whatever. They did get a lot of support from America. That whole political area is very fraught and it's still not over, from the sounds of things.
I grew up in Canada, have lived in Ireland and now live in England. My mother is Finnish (and we used to spend summers there when I was a kid) and my father is Canadian of English-Irish extraction, mainly. I have always considered myself Canadian. I grew up there, lived there till I was 23 and identify more with Canada than any other country. I do tell people "I'm half Finnish" or "a lot of my ancestors came from England" or whatever. But I would never call myself "Finnish" or "English". And DEFINITELY not Irish The thing about my Irish connection is, it's largely Anglo-Irish...some of my Yorkshire ancestors emigrated there, my grandfather was born in Dublin and educated in England and eventually moved to Canada. I never knew him, sadly. He would definitely have been considered Anglo-Irish but apparently he eventually considered himself Canadian.
I will admit to having some romantic ideas about ancestry, though I wouldn't share them with everyone. I guess they are sort of Jungian and mainly artistic/aesthetic/emotional more than anything. Basically, I think it is possible to feel strong and unusual and otherwise inexplicable connections with a country that your ancestors come from. After living in Ireland for three years, as much as I like the Irish and have/had some good Irish friends, I felt that I was anything but "Irish". I just don't think the national character/temperament reflects mine hardly at all. In fact, I feel like I have very mixed "Anglo-Irish" feelings about the place! On the other hand, I feel a very strong connection to England and think I may end up staying here long-term (though the thing is, I live in London, and it's London more than anywhere else I feel that connection to - and London is not England!). A lot of my ancestors are definitely from here. I don't know, maybe it's just romanticism. I still love Finland but since my grandmother died more than fifteen years ago I've only been once. I do feel a connection, of course, but it's more of a childhood connection. I never learned the language properly. And while my mom is far more Finnish than Canadian (she does not consider herself Canadian at all though she's now lived there the majority of her life), I don't think she identifies all that strongly with Finland either. She's kind of her own nationality.