It only makes sense that Notre Dame would be associated with the Irish being that they are a Catholic University located in Indiana. Unlike the Italians, a large number of Irish ( like the Germans ) did not stay on the east coast but migrated to the farmlands of the midwest. The Irish were without a doubt the predominant Catholic presence in that part of the country. So it makes sense that a Catholic University would be associated with the majority of the population of the same faith.
The only thing in question would be the " fighting" part of the name. The Irish, exclusive of their military involvement, were always known as fighters. Their numbers dominated the boxing ranks and they were known to mix it up at the drop of a hat. They prided themselves on that fact. I doubt that the PC motivated Wikepedia would have even considered that aspect.
While not coming from Irish stock myself, I became immersed in their cultural history for both the midwest and the east by association. My first girlfriend was Irish and I sort of was adopted by the family so that relationship lasted longer that than the one with the girl. I had a key to their house and till their death was treated as the son they never had.
The mother was from the east coast and the father "himself" was from the midwest. The real character was granny who was in her ninety's and if she liked you, she would challenge you to put up your dukes and fight her. She came from a family of fighters and I was told that her oldest brother once whipped John L. Sullivan in a street fight. I sort of questioned that until I read that J. L. Sullivan often spoke of that very incident himself. So I can truly say - I hugged the person, who hugged the person, who whipped John L Sullivan. This sort of cultural dynamic more than likely was where the "fighting Irish" came from. Although now it is something looked down on, then it was part of their cultural values and they took pride in being able to fight.