Many are on Medicare. They have no choice giving their SS# to the Dr. because their Medicare number is same as SS#.
I tried that in about 1992...But when I got to my referral specialist...they ALREADY had it...I have tried not to since....but it’s a battle
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.