Per this FAQ on SSNs, it appears that while the government is restricted as to when it may ask for your SSN, private companies, for the most part, aren’t:
All HIPAA does as far as the privacy of your SSN is restrict the pharmacy/doctor/whatever from releasing your social security number and other identifying info, except they can give identifying info, such as SSN, when turning you over to a collection agency.
Under HIPAA, a covered entity is prohibited from sharing your information, including SSN, with anyone. This does not mean that some scuzzball employee won’t steal your SSN and use it for nefarious purposes, but the chances of that happening are pretty small.
If this is a concern to you, you might want to ask why they want your SSN, because unless you’re paying them directly, the only possible need I can see is if they use it to cross-check it against state Medicaid/Medicare/CHIP rolls.
My insurance company ID IS my SS number. Perhaps that is why they require it, because it is also your subscriber number. Although I see you say discounted, not insurance. In that case, why not just make one up. I’m sure their reason is innocent, but if that is the case then making one up will not hurt anyone.
It’s never a good idea to lie to your insurance company, as it’s a really easy way to lose your coverage. Perhaps the OP should follow jasg’s example and merely request that a different number be used.