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Emotional Support Animals is a term commonly used in USA, however they are not legally recognised in Australia. However you may choose to get a letter from a mental health professional to ask if your emotional support animal can remain with you in situations such as housing, but legally there is nothing that can make a housing owner allow an emotional support animal.
This website explains it quite well:
Emotional support dog Australia
Are Emotional Support Animals And Assistance Dogs The Same?
An assistance dog is a dog that has gone through EXTENSIVE training to assist a person with a disability. Their rights and roles are defined by the Disability Discrimination Act 1992.
Assistance animals are not pets. Rather, they help a person with a disability to participate in everyday activities. For people with sight conditions, these dogs are often called “guide dogs.” But, assistance animals can be trained for other roles, too. For instance, they can aid people with hearing impairments, physical disabilities, medical conditions such as epilepsy or diabetes, as well as mental disorders such as anxiety, PTSD and suicidal attempts. They also need to meet certain hygiene and behavioural standards.
Emotional support animals aren’t recognized by Australian law, so the definition is rather vague. These animals provide emotional support to a person through different mental conditions, but they’re not trained to do that. Instead, their pure presence provides comfort and company.
In practice, assistance and emotional support animals differ by their public access rights. This means that assistance animals can go to different public places and forms of transportation. Their owners need their aid in these places and denying them these rights would be discrimination.
Owners of emotional support animals, on the other hand, don’t have such rights. An emotional support dog is considered just a companion dog in Australia. And while the support they provide can be crucial for a person’s ability to properly function from day to day, they’re not trained for the job. For that reason, they’re not granted access to public spaces and transport.