SSN is, unfortunately, a big deal. With your SSN, date of birth, and name, anyone can open lines of credit in your name. This includes credit cards, loans, etc. I found this out the hard way when someone opened a credit card in my name after somehow obtaining my name, DOB, SSN, and home address. (I never found out how they got this information and never will.)
I was lucky in that the thieves paid for rush delivery of the card and the card was sent out before their address change request (from my address - needed to open the account - to their address) went through. The card arrived at my house and I was able to close the account and freeze my credit to make sure this didn't happen again. Had the card gone to them, though, they would have run it to its limit (which was more than $5,000) and not paid the bill. I would have found out about it when the collection agencies banged on my door for payment and they wouldn't be likely to take "But I didn't open that account or spend that money" as an excuse for not paying "my" debts.
So threatening that you'll reveal someone's SSN is a very big deal - especially if you link it to their name and DOB.