They do have to accommodate ADA service animals, but a service animal must be kept by a person with an ADA recognized disability (which would mean at very least more than self diagnosis with anxiety) and the animal must be specifically trained to assist the disabled person with their disability.

The issue gets dicey because a property owner is limited in what they can demand of a person with a service animal. It doesn't have to be registered or licensed for the service. You can't require registration of an ADA disability. So you can just lie and they can't do anything unless the animal becomes disruptive.

Also, airlines are very reticent to generate bad press over that.