While Notre Dame may not have the level of intentional, committed Catholics of a place like Stubenville, it is still determined to be identifiably “Catholic”. As opposed to many top universities e.g. Harvard, Duke, Northwestern, etc that started out as religious but have long since dropped any formal affiliation. And there’s not a requirement that you be Catholic to be a student - remember Raghib Ishmael? Many students choose to go there because they will get a good education with a good name behind it. By academic reputation, it is one of the top private universities in the Midwest, religious or not.
They also don’t require the faculty members to be Catholic. At least in “non-religious” fields like sciences and engineering, they want top scholars, who at the very least are not antagonistic to the notion of a Catholic university. If they put a requirement on all of their faculty to be practicing Catholics, they would eliminate 90% of the top talent from consideration. This is actually a big problem for them I think. At the elite levels of some academic fields, just the very notion of any sort of religious university, let alone a “Catholic” one, turns people off. Even among the very bright, there is sometimes a kneejerk reaction that “Catholic” = repressive, unenlightened, etc. So a school like Notre Dame that wants to be considered “elite” is automatically cut off from competing for good faculty members who would happily consider a secular private or public university. And as it is, the location (cold winters, not on the East or West coast) drives away at least some fraction of the faculty talent pool anyway. What this means for the leadership of the university is that there is a tension between remaining proudly “Catholic”, and keeping their top-tier ranking.
As for actually being Catholic… on my one visit there, I was impressed with the number of people that showed up for daily mass at the basilica. And the mass was done in a revererent manner, without silliness that you sometimes see in university chapels. I also know multiple graduates of ND, who came out all over the board, from being faithful practicing Catholics to wanting nothing to do with the church. But in that sense, you get out of it what you put into it I suppose.