Over the last few pages, there's quite a number of posts all claiming that the people of the Republic of Ireland understand the "I'm Irish!" declaration from Biden and others like him as referring to ancestry.
This is treating the people as though they all have the same view. We don't really have to go far to find anecdotal examples of how there are Irish people who correct Americans on this:
“I’m 100 per cent Irish,” says Brian Kelleher. His paternal grandparents came from Galway. He’s puzzled, though. “A friend of mine has been to Ireland, and he was in a pub somewhere, and he said he was Irish. He was gently corrected and told, ‘You’re Irish-American. We’re Irish’.”
And:
“I think the Irish in Ireland see us as Americans, the high and mighty Americans, who are well-to-do, and who made it, especially here in Boston,” says Kim Camillo."
Mary Beth Keane had an article in the Irish Times with the headline "‘I am so sick of Irish-Americans wishing for an old country they’ve heard about in stories’". She wrote:
"...because I know America has been good to us, and it would be wiser not to deny what I actually am: American. That's how our Irish relations see us, so why are we so reluctant to see ourselves that way?"
Already we can see that Americans of Irish ancestry have had what we can correctly describe as mixed experiences when declaring "I'm Irish!" to people from the Republic of Ireland. The same holds true for those of us who are from Northern Ireland too.
So, it's clearly not solely a British thing. People from the UK and Republic of Ireland have the same understanding of the English language. They both make indistinguishable distinctions of language identifiers, i.e., no real difference at a fundamental level.
And people from both are more often puzzled by such declarations as well as viewing them as a source of amusement to some degree.