I don’t think so. Many people still form a strong identification with the country and culture of their ancestors that last for generations. This may be less so for the British in the New World, where their very Britishness came to define the dominant culture, but for minorities—Irish, Italians, French, Chinese and Indians too—this is often very much the case.
That being said, it would seem that Donahue was hamming for the camera in this case, playing up the stereotype of the ‘Fighting Irish’ so to speak.
All the same, there is something very real about his ‘Irishness’ too. it is not like the cultural influence just vanishes. It transforms in different ways that the Irishness of those from the homeland for sure, but no Europeans in the New Country have exactly ‘gone native’.
A linguistic anthropoligist once brought up the idea that the English of Americans likely sounds closer to the English of the seventeenth century Englishman than the English of today’s Londoner. Americans are more conservative after all, and may well have held onto the ways of their ancestors more than the more liberal Englishman on the British Isles.
Culturally speaking, bill Donahue may well have kept as much of his Irish culture as today’s average Dubliner too. A typical modern Irishman may not like what he has to say, but it is very possible that an Irishman from the Eastern Seaboard is closer to his Irish roots than might be a citizen of the Emerald Isle.