A site can very reliably identify you (across browsers, even) if it knows your public and private IP addresses. Your public IP address identifies which organization or network you're in and your private IP address identifies which machine within that organization or network you are using.

Local IP will remain consistent across sessions and so can be used to identify you as certain device at a given public IP.

They also see my IPv6 address and all of that make fingerprinting more accurate. Real world example https://webrtchacks.com/dear-ny-times/