A Delta spokesperson confirmed the airlines' policy with Tech Insider:
Delta complies with the Air Carrier Access Act by allowing customers traveling with emotional support animals or psychiatric service animals to travel without charge in the cabin. ... We review each case and make every effort to accommodate our customers' travel needs while also taking into consideration the health and safety of other passengers.
Although other animals on planes have caused quite a stir, there's no sign that this turkey was any trouble for the crew or passengers, who were probably glad they had something to watch other than the in-flight entertainment.
As for the turkey? He soared higher in one day than most do in their entire lives, and he never even needed to lift a feather.