There is no registering body for either service animals or emotional support animals in the United States. The registry sites that you mention, and any other website that claims to register an animal as a service animal or an emotional support animal are scams, and are held in contempt by those in the service dog industry.
is correct- emotional support animals are different from service animals, and emotional support animals are not granted "public access' (ie, able to accompany their handler into non-pet-friendly places) like service animals. Emotional support animals are allowed to fly in the cabin of planes with their disabled human, and are also granted protection in most housing. In other areas, they are looked at legally the same as a pet.
The reason that they have these protections but not public access is because of the laws that govern each of these areas: The Americans with Disabilities Act or ADA (which is what grants service animals public access), The Air Carrier Access Act or ACAA, and The Fair Housing Act or FHA. Before I go into detail on the laws, however, I'll elaborate on the difference between a "service animal" and an "emotional support animal" (which I'll abbreviate from here on out as ESA).
First, it is important to note that the only law I mentioned above that makes a distinction between service animals and ESAs is the ADA. The ACAA and FHA do not. The reason that the ADA makes a distinction is because when the law was first put into effect it did not, and there were a great many cases of people using questionable species as service animals (such as snakes, rats, parrots, monkeys, etc) or using dogs that displayed troubling behaviors (such as aggression towards certain people/other dogs/children/etc). The way the law was written, it was made to be reviewed within 10 years of first being enacted. When it was reviewed, it was changed, and in 2011 the distinction was made between ESAs and service animals.
Now, a service animal is defined under the ADA as "an animal specifically trained in work or tasks to mitigate the handler's disability", and under the ADA a service animal may only be a dog or miniature horse. The ADA mentions ESAs only to state that a dog whose sole purpose is to provide emotional support is not a service animal. There may actually be a service animal whose main tasks involve emotional support, but the distinction is that they have had some form of training in this area. What the ADA requires for a service animal legally and the most ethical way to use/train a service animal are slightly different, though. I highly suggest International Association of Assistance Dog Partners
for more information about the ethical use of a service animal. I'd also be happy answer any other questions you have/clarify anything I've been unclear about.
A service animal is allowed by law to accompany its disabled handler into any place of business that the general public is allowed so long as its presence does not represent a fundamental alteration to that business' services. Very few things would actually be upheld by the courts to be a "fundamental alteration".
Here is a link to the ADA revision made to the policies surrounding service animals/ESAs: https://www.ada.gov/service_animals_2010.htm
It is important to note that proper training of a service animal takes upwards of 2 years, is difficult, often costly, and a great many of the dogs who start training as service animals do not finish the training due to unsuitability to the job, even those from purpose bred lines and professionally trained by programs dedicated to producing service dogs. Most dogs would not make a good service dog candidate.
I talked a bit about the ADA already, but I will elaborate a bit more below. ADA
There are 3 titles, or sections, that the ADA covers.
Title I- Employment
Title II- State and Local Government
Title III- Public Accommodations
Public access with service animals would fall under Title III. Asking for a reasonable accommodation to bring your service animal to work with you would fall under Title I. Bringing a service animal into a state courthouse, or living in a state funded housing complex, would fall under Title II.
Frequently asked questions page about service animals: https://www.ada.gov/regs2010/service_animal_qa.html
Homepage about the ADA: https://www.ada.gov/archive/qasrvc.htm Fair Housing Act
The Fair Housing Act does not speak specifically about "service animals" or "emotional support animals" or make any distinctions of training or species. The FHA talks only about "assistance animals". The FHA does not require an assistance animal be any specific species (ie, it could be a ferret or rat or cat as much as a dog), and does not require it have any training. Under the FHA, fully trained service animals, animals in training to be service animals ("service dogs in training", for example), and ESAs are all under the blanket of being an "assistance animal". When you're requesting an allowance be made in housing, you need to request an "reasonable accommodation" from your landlord or housing board or apartment complex. This is where a doctor's note may be required by to verify your need for said accommodation (I will talk more on this below).
Here's a good rundown on that: The Fair Housing Act (FHA) and Assistance Animals : The Humane Society of the United States
The federal site that talks about the FHA: Fair Housing -- It's Your Right - HUD Air Carrier Access Act
Again, no distinction is made between ESAs and service animals. Some airlines do do ban certain species (such as rabbits and ferrets) from flying. Most airlines I know of will say you need a letter of need from a medical professional written in the last year for an ESA, or for a service animal for a non-physical disability, though they don't always actually ask for it. I can go into more detail on this and provide links to it if you would like, but figured I'd just touch on it for now. "THE LETTER"
A letter from a medical practitioner is referred to as a letter of need. The letter that these registries give you technically fulfills this requirement in some cases, since they often employ doctors to write it on their letterhead after a short "consultation" (usually an on-line questionnaire of some kind, I think). A letter of need should be written by a medical professional that is currently seeing you for the disabling condition(s) in question to be of optimal efficiency/use. Pretty much the only times you would actually need said letter would be: when applying to a housing authority/landlord for a reasonable accommodation for an ESA in housing or possibly when flying. With an ESA, you'd not be going into non pet friendly places with it, and with a service animal it would be inappropriate to be showing a doctor's letter, since it is not required by law and the establishment is then likely to think the next service dog team "should" have one to show them.