Remember that Verizon is not a GSM network or are their phones (though some have both radios in them, like the iPhones and the Blackberries), they're a CDMA phone network.
The SIM card for their phones is not the same as a SIM card for a GSM type phone on a GSM network.
It's been a few years since I was on Verizon, but if I remember correctly, their "sim" card was to enable LTE. They have to specifically activate and deactivate their phones on their networks.
The Verizon phone that I had a few years ago, when I was with Verizon, didn't need their "sim" card to work. It would work fine without it, just no LTE, only 3G.
So it sounds to me like the person who sold you the phone, never de-activated it by calling Verizon and you also, have not de-activated your old phone and activated the new phone that you just got.
The new phone is still on the other persons plan.

Fred