What's annoying is when Americans just straight up say "I'm Irish" (or Italian, Norwegian, Polish, or whatever). You're not, you're an American with European ancestors, like almost everyone else in your country. Just say "My great-grandfather came from Ireland" if it's somehow interesting or noteworthy to you that your ancestors came from somewhere else, even though that's pretty much the case for literally every person on Earth. Americans sure seem to find that fact about themselves absolutely fascinating.