The gun is his, for all practical purposes, and there are no registrations required.
If it should ever be used to commit a crime, the serial # may be traced back to the original purchaser, through the gun store records of the place that the original owner made the purchase...if it was purchased recently enough for there to be any records. Beyond that, an investigator would have to track it down by interviewing someone who knew where the gun went, when the original purchaser lost possession of it.
Basically, it's like inheriting a toaster. Nobody really cares.