No, an "Emotional Support Dog" is not officially a Service Animal. As a private owner, you do not have to allow Emotional Support dogs on your property. Even Airbnb's policies state that you do not have to accept service dogs if it causes you or your family medical harm (like being allergic).
The law that allows a trained service dog to accompany a person with a disability is the Americans with Disabilities Act (ADA). An emotional support animal is an animal (typically a dog or cat though this can include other species) that provides a therapeutic benefit to its owner through companionship.
Service animals are defined as dogs that are individually trained to do work or perform tasks for people with disabilities. These tasks can include things like pulling a wheelchair, guiding a person who is visually impaired, alerting a person who is having a seizure, or even calming a person who suffers from Post-Traumatic Stress Disorder. The tasks a service dog can perform are not limited to this list. However, the work or task a service dog does must be directly related to the person's disability. Service dogs may accompany persons with disabilities into places that the public normally goes. This includes state and local government buildings, businesses open to the public, public transportation, and non-profit organizations open to the public. The law that allows a trained service dog to accompany a person with a disability is the Americans with Disabilities Act (ADA).
An emotional support animal is an animal (typically a dog or cat though this can include other species) that provides a therapeutic benefit to its owner through companionship. The animal provides emotional support and comfort to individuals with psychiatric disabilities and other mental impairments. The animal is not specifically trained to perform tasks for a person who suffers from emotional disabilities. Unlike a service animal, an emotional support animal is not granted access to places of public accommodation. Under the federal Fair Housing Act (FHA), an emotional support animal is viewed as a "reasonable accommodation" in a housing unit that has a "no pets" rule for its residents.
The Fair Housing Act (FHA) does apply to almost all housing types including those for sale or rent. This includes apartments, condominiums, and single family homes. There are some major exceptions, such as buildings with four or fewer units where the landlord lives in one of the units. The law also excludes private owners who do not own more than three single family homes, do not use real estate agents or brokers, and do not engage in discriminatory advertising practices. For more on what the FHAct covers, go to http://portal.hud.gov/hudportal/HUD?src=/program_offices/fair_housing_equal_opp/FHLaws/yourrights
https://www.animallaw.info/article/faqs-emotional-support-animals