Emotional Support Animals are NOT service animals.
Service dogs (and in a few cases those miniature ponies are trained as service animals) are NOT pets. They have to be specifically trained in at least one task to aid a disabled person. "He cuddles with me when I am upset" does not count as such a task. "She enters my apartment before I do and looks for the presence of others " or "He keeps people at a specific distance from me during a meltdown" or "She helps keep me balanced instead of suprapronating so I don't fall" might.
An Emotional Support Animal is a pet. They do not require training in at least one task to help a disabled person.
Yes, a bird or birds can function as an Emotional Support Animal. I know of one person who has a peacock for their Emotional Support Animal.
Reptiles, frogs, fish...as well as cats and dogs have all been Emotional Support Animals.
In the USA, an Emotional Support Animal is allowed to live in an apartment even if the landlord does not want them to. An ESA may NOT accompany an owner out in public like work or restaurants (unless the work or restaurant allows PETS). An ESA has certain limited rights to accompany someone on a plane.
Therapy dogs have to take classes and pass a test in order to be allowed to visit patients in hospitals, hospices, clinics or other settings. They cannot whine if separated from their owner by a closed door. They have to accept the presence of durable medical equipment. They have to have a certain temperment as well as training to become Therapy dogs.
What you actually want is a bird or birds that can function as an Emotional Support Animal. This is something that can happen for you if you go through the proper channels-- having a social worker, psychiatrist, psychologist, or other mental health worker evaluate you to ensure that you would benefit from an ESA, then write it up as part of your plan, and then issue a letter that will give you and your ESA a few protections under the law. [A landlord who charges extra for regular pets is not allowed to charge you extra money for your ESA for example].
If you or the people you live with own the place you live at and you are not taking your pet on a plane, then you do not have to declare them to be an ESA.