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Well, yes, and if one continues that line of inquiry further one might argue that things like due process and informal conflict resolution mechanisms are a good idea in employer-employee relationships, just because of the outsize power differential.

The point is, in the context of law, you can't just advocate for an agreement over a gentleman's handshake: you have to be in a position to enforce it. And enforcing restrictions on speech of this sort affects far more than just college professors with unpopular opinions. The 1st Amendment also covers anonymous speech.

Does it or another part of the US constitution protect their anonymity?

Politics is just decision making in groups of people. Similar analyses can be used whether you're talking about government, schools, your workplace, open-source projects, your household, etc.

As a general rule I tend to always interpret these discussions as "how humans interact with each other" unless the discussion itself is about the law.