I wonder if somewhere along the line a sale went south and the seller withheld the airworthiness certificate but the buyer just decided to fly it and pass the paperwork problem along. From a few searches this plane was issued the AWC on "1962-05-21" so its pretty funny that Cessna can't dig up that same number.
More likely, it just got lost. Many owners have no clue where the required paperwork is for their airplane. Pretty much every ferry job I’ve been part of had screwed up or missing paperwork that wouldn’t have been corrected had I not insisted it be right before I came to get the plane.