Westboro is not affiliated with any Bsptist organization that I am aware of. Not sure if Notre Dame is affiliated with the Catholic church or not.
They're technically able to call themselves a "Catholic" University because the Bishop of South Bend/Ft. Wayne has not revoked this right. But when then-Bishop John D'Arcy was challenged on this in 2009, he did boycott ND's commencement and went instead to an alternative commencement organized by "ND Response" (a campus pro-life group) and attended by an estimated 3,000 people.
But he would not revoke ND's right to call themselves a "Catholic" institution.
This ambiguity about "Who's really in charge here" is very intentional, I think. It gives ND tremendous institutional "cover" to do whatever they want.
Disturbing lack of leadership on the part of our bishops.