Anecdote: I've been told previously that my state kept vaccine records in a shared database indexed by SSN--as a way of easing loss of medical records when patience switched providers.
It's possible they were doing something similar.
I've also had to give my SSN to verify my identity to my insurance carrier at my dentist.
SSN is the de facto userID for lots of government systems.
It’s sad for lots of reasons, but understandable considering that it is useful. for whatever reason there isn’t much political will to make an honest effort at a federal government ID (even though we de facto have multiple)
It's possible they were doing something similar.
I've also had to give my SSN to verify my identity to my insurance carrier at my dentist.