I was on a UA flight a week ago where the passenger took his "emotional support animal" out of the small carrier (fit under seat). He was asked twice to put it back in because, while the animal can ride in the cabin, it cannot be removed from the carrier. He refused both times until the Captain came on the PA to let the whole cabin know that he was turning the plane back to Miami (we were over the Gulf enroute to Costa Rica) if the passenger did not comply. After a few scowling looks, he did it.
That wasn't an emotional support animal, because emotional support animals aren't carried in pet carriers. If it was an emotional support animal, then the passenger could hold the animal on his lap or the animal could sit on the floor. If the pet was in a carrier, then it was probably a cabin pet, and the rules prohibit removing a pet from its carrier.