I went to a dealer to buy a car in cash (well, a cashier's check). They had a touch-screen computer with all their forms for me to fill out. It came to a credit approval/application form, and I refused to fill it out because I wasn't opening a loan. The lady assured me it was "standard" and that it wouldn't be used for anything, because I didn't need the loan/credit. I kept refusing, and she kept insisting they only needed it because the software required it to continue. After this impasse, I stood to walk out, and she suddenly changed her tune and somehow "figured out" how to bypass the form. Classy.
That's why they had a cashier's check and not a personal check.
Because, considering it sounds like you bought the car outright, there's absolutely no reason for that form to need signing.
> You are not required to have a social security number to open a checking or savings account.
Source: https://www.consumerfinance.gov/ask-cfpb/can-i-get-a-checkin...
Having spent more than a few years in FinTech, I know these things can vary from company to company as companies generally are required to design their own “risk-based” policies for AML and KYC.
(I'm not undocumented, but I imagine they see that enough which is why they don't bother saying anything when I leave it blank.)
I'm pretty sure they just want your SSN for collections/credit-reporting purposes.
No you do not.
Anecdotal: I always leave that field blank and have never been asked for it.
Anyone can ask you for anything, anywhere. It doesn't mean they actually need it, or that you have to give it to them.