Why am I being asked for my full social security number to activate replacement phone?

I am trying to activate the phone sent to me to replace my broken phone and the chat agent redirected me to a "secure" link requesting my full social security number and birth date. He told me to leave the window open when I had entered the information and return to the chat. I ended the chat and will go to the store in person tomorrow. If I hadn't had the experience below I might have gone ahead and provided this information.

Two weeks ago I called Xfinity about my internet service, clicking on the phone number linked to the site, and the agent was asking me for the numbers on the back of my router. When I turned around and looked at my laptop (he had me log in to my account), someone was trying to buy Amazon gift cards on my Amazon account. At the start of the call this person knew the balances on both my mobile and internet accounts, so they definitely either work for Xfinity or have access to account information.

What is going on? I have zero faith now in the security of my personal information with Xfinity, and and left with no options other than in person interactions at a store.