Actually, I don't think that is correct. As a rule, emotional support animals are not trained to be anything. The Doctor simply needs to put in writing the statement that the animal is needed by the affected person. This means that Fifi who has never been trained to do anything other than defecate in the yard and not on the living room carpet can be "certified" as an ESAN. Yes, service animals are trained to be compliant. ESANs (Emotional Support Animal) are not classified as service animals the way that say an assist animal for a blind person is. On the passenger manifest, there are separate listings for ESANs and SVANs (service animal).
I was told by a gate agent that there is no prior investigation by the airline to determine that the person writing the ESAN statement is even a doctor! It's such a touchy subject, and the airlines fear being sued. As a flight attendant, I am supposed to verify that any person bringing an animal onboard is listed on the passenger manifest as having either a service animal or an ESAN, OR has paid the fee ($150. I think) to bring a pet on board. Because, if Fifi is just a plain vanilla pet, the airline is held (at least, partially) responsible if Fifi bites another passenger or animal. In the fee paid situation, the animal must remain in a closed pet carrier at all times while on board, and no I don't care that Fifi is scared of the engine noise, you may not hold her in flight. Fifi stays in the carrier and the carrier stays under the seat in front of you--end of discussion. (Which also means that passengers with "just a" pets may not be seated in a bulkhead row.)
(You may or may not be surprised at how many people hide a small dog, cat or other pet in one of their carryon bags to avoid having to pay the fee for taking the animal on board. We caught one several years ago because the sneaky passenger with a cat in her purse, got seated next to a passenger who was highly allergic to cats.) Here's the catch...if the passenger uses the words "service animal" the verification process stops immediately. Now, I still have to write a report that I saw no documentation of service animal status--ostensibly so that the company can investigate further. But, I can not pursue the issue any further if those magic words are said.