America is multicultural with a lot of diversity among white Americans that perhaps isn't obvious to British people without a strong sense of immigrant America. Joe Biden went to Catholic school which means fee-paying in the US - but they were generally 'cheap' private schools run by religious orders, very different to the elite prep schools.
His children were christened and raised as Catholic, so it's something he has sought to pass on. US Catholics tended to sort themselves into parishes they felt more comfortable in, usually on ethnic/national lines and following patterns of migrant settlement. So for Biden to say 'Irish' is a shorthand for a particular American way of life which is basically religious, Catholic, urban - but a specific version of that, the Irish American version.
Someone said Biden is an English surname - my guess would be they were Protestant but in a mixed marriage the Catholicism got passed on. There weren't historically so many English Catholics migrating to the US. I could be wrong. It seems though that the Irish identity survived in his family more than the English. Other families did differently.
The fact people don't explicitly say 'British American' doesn't mean that Britishness/Englishness hasn't left a cultural legacy. And WASP was more salient as an identity earlier last century. Remember Larry David pretending to be a WASP in Curb to get into a country club - 'I'm a Moose. And an Elk'. The Mid-West has a strong Germany heritage, though people don't often define as German-American - the wars made a difference there. But American Christmas & a good deal of the classical music world has been influenced by German American cultural practice.
What counts - only linguistic or physical difference? Why should somebody on the internet get to say over the person themselves? What would you count as authentic? Biden is attached to Irish poetry and history, visits frequently, has built friendships and connections with extended family and takes a special interest in the region of one of his ancestors. Why not do this if you could?
There are times when it would be good not to recycle tropes, whatever the satire/intended meaning/valid criticism blah blah blah justification after the fact. A little bit of political stress and whooops, the political cartoons start to get a bit tropey. There have been too many examples over the past few years. I guess there might be a perception among some British that this is just punching up, because 'America'. But it isn't, and it reflects badly on British political culture.