All cellular Apple Watches use an eSIM. There is not physical SIM card.
First, does your Apple Watch Series 3 have a Red Dot 🔴 on the Digital Crown? If it does NOT, then you do not have a cellular radio in your Apple Watch. If there is a Red Dot, you have cellular radio, and can set up cellular service
The Cellular Apple Watch shares the iPhone's phone number.
So Verizon should be talking about setting up your Apple Watch with your iPhone's phone number.
Do any of these articles help
https://www.verizonwireless.com/support/knowledge-base-215386/
https://www.verizonwireless.com/support/knowledge-base-215347/
https://www.verizonwireless.com/support/apple-watch-series-4/