They want to be sure you will not enroll others when you open 六分

With SSN, the spouse will be fixed on that spouse.
It is normal they ask SSN for each insured family member.

Later on when they ask you verify the spouse relation by showing your Joined TAX return, the SSN must be correct. Yes, company do that.

Otherwise, when you open  六分店, 七分店, 八分店 ect later, the company will not know who is who...