I care, being a government parasite myself. I retired from a government job and find this more than a little concerning. Of course they ask for your damn ssn# now to sign up for yoga what diff?
Law school administration got a little pissy about it, but eventually gave me their made up number for student ID. They couldn't show me where, in the law, they were entitled to my SSN. It's only purpose was student ID number.
The other guys do it for credit check. Give them a CC number, they can run that.
Sometimes you will have to take your business elsewhere, and saying "no" always increases the time it takes to do business. But that's the price you pay to limit the circulation of your SSN.