Most (all?) bikes have a serial number stamped on the bottom of the 'bottom bracket' the part that the pedals go into. These serial numbers are unique per bike brand, but not universally unique.
The serial is pretty hard to remove, without it's removal being obvious to anyone that looks, though the brand can be removed or changed - if that happens the serial wouldn't show as stolen.
If there was a lookup for serials then you could check if the bike was stolen, or if the bike was incorrectly branded (serial doesn't match brand), but there isn't a central place for this, unless the (ex)owner registers it on a service such as immobilise.com
My fiance's bike was stolen a few months ago. By coincidence someone who was a friend of a friend of her's ended up buying it off of craigslist. We discovered this when he arrived at a BBQ riding her bike. When we asked about the details of his purchase, he sent us all the email correspondance with the bike thief.
The thief had actually given him the serial number of the bike, which he checked in the online police database and it returned no hits. The catch was that the thief slightly altered the serial number so that it wouldn't yield any hits. When he went to actually look at the bike he didn't notice the subtle difference in the serial number and assumed it to be legit.
Motor vehicle VINs in North America use a check digit to detect VIN errors, which consequently makes it hard to modify a VIN without failing checks at time of purchase (such as when the VIN is looked up via CarFax or at the DMV). It's not cryptographically "hard" to modify it, but additional check digits up the difficulty (like appending a hash to the plaintext serial number).
The OCR-A character set used in the MICR numbers on checks makes it easy to spot a modification like the one in your story.
Interesting, I was thinking something that could be read easily an not easily tampered with. Say an RFID embedded serial number in the bike emblem/badge.
The serial is pretty hard to remove, without it's removal being obvious to anyone that looks, though the brand can be removed or changed - if that happens the serial wouldn't show as stolen.
If there was a lookup for serials then you could check if the bike was stolen, or if the bike was incorrectly branded (serial doesn't match brand), but there isn't a central place for this, unless the (ex)owner registers it on a service such as immobilise.com