You sometimes see signs in restaurants prohibiting all animals except service animals. Service animals and emotional support animals are not the same things. Different laws apply to service animals and emotional support animals.
The definition of service animal under the Americans with Disabilities Act is a dog trained to do work or perform tasks for an individual with a disability. A service animal does not require professional training, and no certification is required.
An emotional support animal under fair housing laws can be any animal, not just a dog. No training is needed. All that is needed is a simple letter from a doctor or therapist indicating that the individual has a metal disability and that the animal lessons the effects of the disability.
Note that the emotional support animal rules are under fair housing laws. Fair housing laws are for housing, not places of public accommodation such as restaurants. The rules regarding service animals are under the Americans with Disabilities Act, which does apply to restaurants.