I think there's been a misunderstanding or miscommunication. As a student from the late 90's you likely used your SSN as your student ID and that hasn't been allowed since 2000 or so.

They are most likely asking for your UT EID which, even if you had one back then, you likely wouldn't remember. The university did retroactively create UT EIDs for students going all the way back to the 1960's so you definitely have a UT EID, even if you don't know what it is.

A good place to start would be the UT EID Self-Service Tools where you can use the Find My UT EID feature. Failing that, the UT Service Desk should be able to assist.