It's more complicated than that. If the transaction is over $10,000 it needs to be reported to the IRS. Dealers are often involved on the financing side directly, and their systems are just set up to require a Social Security number to put anything into the system. So because of the financing, some of these "know your customer" rules apply, and they automatically get applied to everyone as universal policy -- employees aren't allowed to use judgement to make personal exceptions. And physical IDs can be forged, whereas the existence of a Social Security number in a database can't be.

SSN's aren't perfect, you can still commit identity fraud. But high value financial transactions get safer the more you can verify about someone.