If you talk to most of my fellow micks, they'll whinge and moan about Yanks 'pretending' to be Irish. They'll take offence to it, like Irishness is little more than a fashion accessory to foreigners. They'll say that such Yanks are annoying and will scold them online for saying "I'm Irish" when they're several generations deep into being American.
Fuck all that noise. If someone wants to be associated with my wee island (and it is mine), I get a warm feeling in my chest like an internal hug. I don't care if the person has never had a single ancestor so much as set foot on Ireland, if they nevertheless wanna be Irish I'm thrilled to hear it. It makes me less self-conscious about my shite accent, it makes me feel like I could have friends in other countries before I even meet them. It's like a cheat code whereby I'm granted, up front and gratis, 30% of the required 'social ingratiation' mileage you usually need to put in before you start getting somewhere with a new friendship. There is literally no downside to it that I can see, and certainly no downside that has ever presented itself to me. The only way I could feel negatively about it is if I already have a casual hatred of Americans, which I don't.
If you don't believe that the cunty "ur not irish m8" attitude is borne of hatred for Americans, show me the reams of similarly-scolded Australians claiming Irishness. Show me the Iraqi with an Irish grandfather being told to shut the fuck up for claiming Irishness. Show me the hoards of Englishmen, even, being told the same. I have only ever seen or heard it directed at Americans.
The Australia example is much more powerful because it's one of the two places Irish people tend to go when they leave the British Isles, the other being the USA. We have been going to Oz in droves - voluntarily or not - for about as long as the USA has existed, so we claim a huge portion of the populace. The USA is still a brand-new country, in the grand scheme of history, so there're legit reasons for people to still feel like their families' previous countries are the more defining ones for their identity. There's nothing wrong with that. Unless you're American, apparently.
In closing, let me talk directly to my fellow spud-botherers: please shut the fuck up. I assure you, between the scolding Irishman and the Irish-loving Yank who never shuts up about it, you are the annoying cunt in this scenario. You make Irish people sound like völkisch wankers obsessed with the purity of their genes and the ethnic make-up of their country. Spend less time gatekeeping national identity and more time on avoiding heart disease, ya greasy hoors.
British Isles was a term pushed by the UK in order to aid their claim to our lands. We are no more part of the british isles than we are the south eatern icelandic archipelago.