Let's say I want to look up who has a particular phone number in my organization, and they're usually listed if you do
finger username. Is it possible to do a reverse lookup and get the username/names that have a phone number in its details?
It should not be possible to do this. IIRC finger looks up details of a user based on looking in that users home directory for certain information. It is not very complex or centralised by today's standards. I guess if you know or brute-force all the users and their details you could build your own database and do the lookup on that.
IMHO finger is a security risk and there is no good reason for running it in the modern world. I've not heard of it for many years, it has virtually no built in security, talks in plain text, allows for system and user probing and wse designed for "a different world" (pre social media, pre ubiquitous cellphones, pre Snowden)