Notre Dame is a Catholic university. Early on it had a student body which was quite comfortable with adopting the nickname "The Fighting Irish."
Then came the Italians (including the legendary football coach, Lombardi). Now ND is a total mix of cultures, religions etc. But they are still "The Fighting Irish."
Here is some history for you. When radio and then television developed the capability to broadcast football games live, but before they could do a lot of games, the execs settled on following ND. As a result, for many folks, ND was "their" team.
Even if they weren't Catholic or had been to college, Notre Dame was it.
That allure still exists today, a generation later.