That watermark

can’t or won’t?

Are you maybe thinking of MAC addresses? That would be closer being the “identity” of a device and you can typically identify the manufacturer from it. You can’t see the MAC address of a remote router via the internet though unless you are on its local network.

An IP address is usually a temporary lease provided by your ISP, and residential connections usually get a new one every once in a while (like every 24 hours).