Medical organizations are now prohibited from requiring your social security number. They can request it but you don't have to give it to them. In the past some insurance providers (e.g., Medicare) used your social security number as your insurance ID (or part of it) but stopped in April 2019. None of them do this any more since it's illegal and puts the patient at risk for identity theft.
By the way, the typical reason they ask for it is so they can better track you down if you don't pay your bill.
I never give my social security number to an organization that requests it, unless they can give me what I consider a valid reason they need it (such as official government organizations like the IRS and Social Security).