Sorry, what is asking for your social security number?? I'm not familiar with the strange practices of US carriers, but I've never heard of anything like this.
What is this new (non-activated) SIM for? I'm never quite sure with VZW because they are a strange hybrid, a CDMA carrier (which doesn't require a SIM) but with LTE (which does), something that's not really found outside North America. But in general the SIM is the thing that identifies your account to the network, and the phone number is associated with the SIM, so changing SIMs will change your phone number unless the SIM is a replacement (in which case you may temporarily have a different number until the carrier's systems update). So why you have this SIM is the key question: without knowing that I can't know what activating the SIM would mean.
But I've never had a phone ask for my equivalent of your social security number, or any personal information of that sort. That's why I don't know what to say to that