No – universities are not fiduciaries*
As mentioned in the answer by user6726, there is no universally accepted definition of what constitutes a fiduciary. However, Lord Justice Millett's explanation in the case of Bristol and West Building Society v Mothew [1998] is helpful (see my added emphasis in bold):
a fiduciary is someone who has undertaken to act for or on behalf of another in a particular manner in circumstances which give rise to a relationship of trust and confidence. The distinguishing obligation of the fiduciary is the obligation of loyalty.
While this is an English case, this description is true for all common-law countries (including the US). The bold appears to be the key point you are missing. Unlike a company director (with a duty of loyalty to shareholders) or a trustee (with a duty of loyalty to all beneficiaries), universities have no such general duty of loyalty.
But they do have other duties
The fact that universities are not fiduciaries does not mean that they have no duty at all. Instead, their duties are contractual, statutory, and (potentially) tortious.
For example in England and Wales:
Contractual: s49 of the Consumer Rights Act 2015 requires contracts for services between businesses and consumers to be performed with reasonable care and skill.
Statutory: The Higher Education Act 2004 created a body for addressing students' complaints which universities will be legally bound to deal with.
Tort: In University of Bristol v Abrahart [2024] the High Court acknowledged controversy as to the extent of universities' duty of care in tort law and refused to comment on the issue (as they found that duty had arisen elsewhere). This would be a less clear avenue than the above two routes.
While these examples are jurisdiction-specific, the overriding point remains that there is no fiduciary relationship. Instead, it is necessary to look to other potential duties that may arise (as the examples illustrate).
*occasionally the courts have acknowledged the exceptional existence of a fiduciary relationship (for the purpose of a particular case only) based on the specific facts (eg Kelly v Baker & Braid [2022]). There is no clear reason why universities should be fiduciaries in the case of your question though. In that sense, this caveat is just for completeness.