The answer is … it depends. It depends on the institution and the diocese.
There is only one university in the US under the direct control of the bishops, Catholic University of America in DC. The Church in America financially supports that school. All the parishes in the US even have a special collection each year for CUA.
Other universities do not have a financial relationship with the diocese though they might get some support from a religious order. Notre Dame falls into that category.
All Catholic colleges and universities are supposed to, at a minimum, not teach anything contrary to the faith. But whether they are held to this is up to the Bishop of the diocese. In some cases, the Bishop has told the college it can no longer call itself Catholic. Marist (in NY) and Marymount Manhattan are two of those.
In the material and websites of many “Catholic” schools you will see wording similar to “a liberal arts college in the Catholic
tradition”. Those schools are not claiming any particularlly Catholic doctrinal orientation but likely have some stained glass in the chapel and statues of Mary on the grounds.
Catholic universities that claim to be doctinally oriented to the Church will have a mandatum or similar statement of faithfulness to Church teaching. The Cardinal Newman Society keeps pretty good track of which instituations have this. It is sadly a very small list.