21.5 Requires the holder of a type certificate for any airplane produced after March 1 1979 to issue a AFM. It doesn't mention anything about serial numbers.
21.50 Requires the Type Certificate holder to ensure continued airworthiness to the owner of the airplane so maybe the manuals have to be serial number specific in order for instructions for airworthiness compliance, etc, to be made available to the owner of the plane.
Maybe someone else can chime in with more specific info...
91.9 requires that the appropriate information be in the airplane.
In addition to the references HSDriver provided, paragraphs in Part 23 (and their CAR 3 equivalents) specify the information that the manufacturer is required to provide, possibly in an AFM. With changes made in a model over time, this required information may change from one airplane to another, so a one-size-fits-all AFM for a model often won't work. In reality, an AFM part number generally applies to a range of aircraft serial numbers. If an AFM is lost, you can replace it with an AFM that applies to your range of serial numbers.
Even if the manufacturer made no changes to the model, aftermarket modifications are included as part of an AFM (as supplements, of course), and this is usually what narrows an AFM down to a particular airframe serial number. And if an AFM is lost, it can take substantial effort to reproduce all the required documentation.