"How long after leaving the country do emigrant families go on seeing themselves as Irish, or Irish-American, rather than just identifying with their new abode? To this outsider it appears to go on for multiple generations."

I think a significant factor is how "isolated" (or, to phrase it differently, how strong and self-sufficient) an immigrant community one joins upon arrival in the new country. That, in turn, will influence the degree to which members intermarry/interbreed with folks of other origins.

The late President John Fitzgerald Kennedy was of 100% Irish ancestry, or ethnicity, or whatever you wanna call it, despite his family being so well-established in the US for so many generations. All sixteen of his great-great-grandparents were born in Ireland and had resettled in the Boston area by the time they had children!

Now, that's not just a statistically unlikely family history, it's evidence of an astoundingly tight community; one should easily understand that all its members, throughout all those generations, would consider themselves "Irish-American" (if not simply "Irish") to a much greater extent than someone whose four grandparents might include, say, one born in Ireland, one born in the US of Irish parents, one Italian American, and one US citizen whose family had been on this side of the ocean for so many generations that he doesn't even know how many nationalities are represented in his DNA.

Descendants of the downtrodden Catholic workers and peasants who fled to America tended to stay together in fairly homogeneoius US neighborhoods, and to look to the Church as protector and a focal point for community. Ulster Protestants and the more affluent Anglo-Irish would be much more likely to become assimilated into mainstream US culture, and have their children and grandchildren lose consiousness of any Irish identity relatively quickly.

I remember hearing on the Today Show one St. Patrick's Day morning several years ago that many more current-day Americans of Irish descent are Protestant than Catholic; that's certainly NOT who you see out celebrating, and identifying strongly as Irish, on March 17. I think part of the explanation is that large numbers of Ulster folk came across the ocean a century or more earlier than the famine-driven exodus of the Catholic/Gaelic population of the Irish West, and therefore had several more generations of breeding to get a "head start" on producing descendants in greater numbers. More obviously, identification with the old country is going to be much weaker for those who have to look back a couple of centuries rather than a couple of generations.