I just registered billyjiminy123@gmail in a private browsing session and didn't get prompted. I even sent an email without being prompted for verification.
Google tracks IPs like Amazon tracks shipping addresses. It's kind of hard to deliver what you've requested if they don't have your address.
If you're using Gmail (or sending email to people who use Gmail), Google already knows a ton about you. They can probably guess where you live without knowing your IP. They can certainly guess where you live if you're ever searching for region-specific stores/parks/directions/etc. I suspect if you're browsing through a VPN regularly, Google has enough data to pick you out of the searches coming from the VPN's outbound IPs with fairly high confidence.