I wouldn't say it's overwhelmingly American. I've spent plenty of extended time in other places (England, Australia) and jokes about heritage are virtually universal.

But yes, people don't actually think that dude is claiming to be born in Italy when he says that

I'm sorry, but this just isn't true. Many if not most people on one side are literally saying exactly that, they're misunderstanding completely what a typical American actually implies when they say "I'm X."

understand your point, but I'm not saying it's particularly egregious, just dumb. Saying "I'm Italian" means something, and somebody with no relation to Italy is simply not Italian.

That's absolutely true, but the average American has absolutely no clue that saying "Hey, I'm Italian", or "Irish", or "French" to an actual Italian or Irish or French person would be offensive - in fact it's the opposite, it would only be meant as a innocent piece of small talk. It means only "Oh that's interesting, my ancestors also came from there way back then" - that's all it means. They wouldn't even realize it'd be misunderstood in the same way a native English speaker might not realize saying "So, you're letting me off the hook?" could be misunderstood by someone not as familiar with English (because, on paper - it means something entirely differently than it does in the figurative, colloquial sense.) I've seen people in England react in absolute confusion/disgust when a fat American tourist woman says she needs to "dig something out of my fanny pack", likewise I've seen my English friend make a quick offhand comment about smoking a quick fag and absolutely fucking confusing people when he's visiting America.