According to the airline (and I forget which airline it was) the turkey met all requirements set by the airline for an emotional support animal, so they had no choice but to allow it on the plane.
Emotional support animals can be any species. Obviously there will be some restrictions as you can't just show up with an emotional support rhinoceros and expect to get it on a plane, but there are examples out there of people boarding planes with pigs, turtles, and even some fish in a bowl. Service animals under ADA can only be dogs (or in some cases, miniature horses).
There was a hilarious article not too long ago out of NYC where the author got herself one of those mail order emotional support animal letters and borrowed a series of animals to take around the city with her to see if they'd be allowed into public areas. At various times she entered local public areas accompanied by a turtle, a pig, a (non-venomous) snake, a turkey, and an alpaca. Apparently one restaurant did not allow her to bring in the snake, but they went in everyplace else. She got this close to boarding an Amtrak train with the alpaca, but it got nervous, so she removed it.