99.999% of insurance cards have your SS number.
If you use any insurance card to pay for health care, they have your SS#.
The only way to avoid this is to pay cash before treatment or give them a credit card for payment. We have some younger relatives in great health. When they need some rare medical treatment, they use their credit card or cash.
Producing picture id is sadly necessary. Fraud is very common with multiple people attempting to use the same insurance policy.
Odd. I have never seen an insurance card with my SSN on it.