They want your ID so you will think twice before screwing a .45 in the salesmans ear and tell him to get the hell out of the car.
I work with credit bureaus all the time. No SSN number no credit check. Plus they have to provide the credit bureau with a federally mandated form with your signature and SSN #.
Give them your ID and then go scare the shit out of the saleman. I did it in a new Lightning a while ago. You should have seen the fokker holding onto the oh shit handle !