Related to the OP, is it acceptable to refuse to answer questions from an anonymous caller?
Yes. I’ve done it. Not because I really care giving the information but lihachis.
I hope the answerer was joking. I’m pretty sure that it’s not acceptable to do acceptable things for unacceptable reasons.
As for the question, why would it matter if the asker wants to
remain anonymous? (If you’re not comfortable with it, then the
question becomes whether you have any obligation to talk to them.
If you don’t, then let them play by your rules or not at all.)