Every Kindle has a serial number. It’s used to register a Kindle to your Amazon account. I just looked at mine under Manage Your Devices. Also, the Kindle serial number(s) are encoded in the books you buy for a Kindle. (don’t ask how I know. But, trust me, it’s in there.)
If a Kindle is stolen it should be easily traced when it connects to Amazon’s server. Or when it’s registered to a new Amazon account.
But, does Amazon bother recovering stolen Kindles?