The animal is not designated as an ESA - the PERSON is certified as needing emotional support. Any pet can fulfill that need. The airlines did start restricting the size and number of animals, and what type of animals, but many did allow rabbits and birds and other things.
Now the dog, as a service dog, will have to be certified that it has been trained to provide a service and be with the person who has the disability. That is IF the airlines choose to enforce this change to the rules. An airline may choose to continue the current ESA policy (but I doubt it).