claiming to be Irish

I can speak to this phenomenon a bit. It’s part of what was drilled into us from our families. My father’s maternal grandparents were from Donegal, Ireland. Any time a single person from a Donegal family passed away in the entire city of Philadelphia, whether they were known to my family or not, my father, his brothers, and my grandmother were going to that wake to pay their respects. Once he became an adult, he became a member of the AoH, which is an Irish-American fraternal order. They’d keep some Irish customs alive (and being separated by the ocean, no doubt hallucinate some new ones). For people that are heavily invested in their families, it’s a way of feeling connected to your ancestors. I think leaving was rather traumatic for many people, so I think there is an element of mourning in the connection for some too.

I myself wouldn’t call myself Irish, but I know a great deal about Ireland and I share a deep appreciation for it despite being a Yankee. I get that it’s no doubt annoying when someone who knows nothing of the place they are claiming ownership of says they’re Irish or Italian to someone actually from Ireland or Italy, but at the end of the day I think it comes from a well intentioned place. If my family came to find we weren’t at all Irish by ancestry, I would definitely feel shocked as much of my upbringing was framed by that identity.

Calling people out as "fetish like" for identifying with...anything... is a bad look.

A person's perception of themselves, their identity or self image isn't for you to qualify as being good enough

No not hateful, you're just giving off a weird vibe about it. But you're half way there actually, transform that energy into curiosity.

The two you picked especially have a real fascinating history and I'd encourage you to check it out because both of those groups had a tough time in their early immigration days. They aren't fetishising at all - those communities had to stick together because they weren't exactly welcome, and that mentality became ingrained. Over time, it was less necessary for survival so it transitioned into more of a cultural tradition.