You aren't required to show your social security card to your employer. You aren't even required to have the physical card for almost all purposes. It specifically says on the card, and in other places that it is not to be used for identification.
> It specifically says on the card, and in other places that it is not to be used for identification.
The US tried that back when Social Security Numbers were introduced. It specifically said it was for tax-purposes (a context where it might've been adequately-secure) and not to be used for anything else.
Yet without any actual penalties against "other places", it got misused everywhere by companies trying to save a buck on primary-key choice and authenticating people.
https://www.ssa.gov/ssnumber/assets/EN-05-10553.pdf