Without saying it, you're essentially explaining that, if OPs curiosity gets the best of them, they then willfully provide personal verification details to a third party without any way of verifying they are authentically them and this isn't a phishing attempt for future identity theft or a scam.

OP, that's categorically a Bad Idea™.

If you feel in your gut that something felt weird about that interaction, trust your gut the first time. If they're authentic, they will provide ways to verify they are indeed who they are. Otherwise, literally anyone can say they're from any company and possibly be spoofing their number on Caller ID while doing so.