A public school (including a university as part of the State) under a governmental entity makes it exempt under section 115, but that doesn't preclude them from obtaining a 501(c)(3) status for individual reporting needs. Otherwise, their reporting is to their school system or to their "owner" such as County or State or School District.

https://www.law.cornell.edu/uscode/text/26/115

I remember we went around and around on this issue in the previous tax year:

https://proconnect.intuit.com/community/proseries-tax-discussions/discussion/mandatory-efile-form-99...