There are several problems there:

• Stereotypically, the Americans doing this are way further removed from their ancestry than the second-generation immigrants you describe (in fact it's completely normal and accepted for second-gen immigrants to identify as their parent's nationality as well in Europe);
• "I'm Italian" and "I have Italian ancestry" are NOT the same sentence. You seem to realize that, but many Americans don't, and the comment you replied to complained about the former, and the difference is fundamental;
• Europeans are generally not on board with the whole "ethnic identity" stuff that Americans do, for a variety of reasons that one could simplify down to "last time we did that, nazism happened". The mainstream progressive view is humanist and intentionally colorblind, and it is therefore profoundly shocking to see Americans derive a sense of self-worth from their blood, because these are the talking points we normally only hear in documentaries about Mussolini...
  Now I have spent enough time reading about how American view their complicated relationship to race, ethnicity, and ancestry, to understand where you're coming from, but this is fundamentally at odds to the humanist approach of "we're all the same and who your great-grandparents were does not define who you are in any way". (Which is obviously idealist, and does tend to "whitewash" some struggles, but it is nonetheless the prevailing approach).