Which breeds are normally chosen for this role, and why?
As @Keshlam states in the comments, there's no way that Stack Exchange can answer this question definitively. In general, I feel the best answer here might be to begin going through the process and see what becomes of it.
In my particular location, there is not a list of "qualifications", and one can only find out if their dog is eligible by beginning the process ; the reasoning for this is that there are too many people who want the Working Dog title so their dog can be with them where otherwise not be allowed.
No special training is required beyond basic obedience, and pretty much any breed can be registered as an emotional support animal as long as the dog meets a few basic behavioral criteria. In fact, you can easily register your dog online.
The National Service Animal Registry states:
According to the Americans with Disabilities Act (ADA), Air Carrier Access Act, and Fair Housing Act, your emotional support animal (ESA) does NOT need any specific training, but only needs to be manageable in public and controlled by you, its disabled handler. This is because the very presence of the animal is what ameliorates the negative symptoms associated with a person's emotional or psychological disability.
That said, emotional support animals are often confused with psychiatric service dogs, who undergo more rigorous certification and are trained to perform specific tasks that their specific disabled human handlers cannot perform. Emotional support animals are not granted special privileges under federal service animal laws in the US, though some states may offer such privileges.
According to servicedogcentral.org,
The difference between emotional support animals and service animals is threefold:
To have a service animal, a person must be so impaired as to have a disability. For example, needing glasses for poor vision is an impairment, but being unable to see with or without glasses is a disability. Having a mental illness is an impairment, but being unable to function on a minimal level because of a mental illness is a disability. Folks may have an emotional support animal due to a mental impairment if they are also otherwise disabled or elderly or they may have an emotional support animal because of a mental illness disability. Only those actually disabled by a psychiatric impairment would qualify to use a psychiatric service dog.
Service animals are task trained to actually do something which mitigates the person's disability. Their defined function is not to provide emotional support (affection on demand or a security blanket) but to do something the handler cannot do for themselves which allows that handler to overcome or ameliorate an inability to perform major life activities. Emotional support animals don't have to be trained, so long as they do not disturb neighbors or pose a threat to public safety.
A person with a disability has a right to be accompanied by a trained service dog which is assisting them in most public accommodations (places of business). A person with an impairment or a disability does not have a right to be accompanied by an emotional support animal unless individual state laws specifically grant this right, in which case it applies only in that state.