I like this idea. But I think it's unlikely to happen. Universities often don't even run their own email anymore (at least for students; faculty often have a local Exchange setup for them). I would have thought that Universities would want to run their own services for file synchronization and sharing (i.e., internal Dropbox), audio/video conferencing (i.e., internal but interoperable Zoom/Skype, perhaps set up with SIP), etc. This would enable them to better manage IP and sensitive student information (SSN's, etc.), but they don't seem to have enough foresight to offer those kinds of services until the faculty are already using a freemium service. Then, _if_ they roll out something, the University version is clumsy and error-ridden.
Additionally, the network effects of other social networks likely means it would be a tall order to convince the sufficiently senior administrator that they should dump twitter and go to their own Mastodon instance.
20 years ago this might've been a possibility. That was back when individual departments ran their own email/web/whatever servers. So individual departments within universities that had more agility and flexibility might've tried something like that. But now most universities seem to be clawing that functionality and freedom away from departments and putting it into a centralized IT department. So email stays because it's already embedded in the culture, but it makes it hard to add something new like a federated social network (or even video conferencing, it seems).