I think you are thinking of form I-9, which is federally required. SS Card is one of may accepted forms of ID, I always use my passport instead.
Using your passport doesn't obviate the requirement that they have your SSN or ITIN for the IRS.
The SSN card proves your eligibility to work, they still need to verify your ID separately (DL, voter card, etc.). They need both pieces. A passport counts as both.
Which is I think what I said. They still need your SSN, just not the card itself.
Are those still as I remember them - with no biometric information (like a photo) and basically no security features?