I don't know where you are, but it sounds like the US because you mention the ADA. In the US, there is no registry for service animals. There is no such thing as a "registered service animal". There is no limitation on what breeds of dogs can be service animals.
Here is the exact wording from
the ADA website:
"Service animals are defined as dogs that are
individually trained to do work or perform tasks for people with disabilities. Examples of such work or tasks include guiding people who are blind, alerting people who are deaf, pulling a wheelchair, alerting and protecting a person who is having a seizure, reminding a person with mental illness to take prescribed medications, calming a person with Post Traumatic Stress Disorder (PTSD) during an anxiety attack, or performing other duties. Service animals are working animals, not pets.
The work or task a dog has been trained to provide must be directly related to the person’s disability. "
(The page also mentions, later, that miniature horses can also be service animals, but let's not even get into that.)
You can train - or "train" - your own service dog to your own satisfaction. The dog is not required to complete any sort of regulated training program. It is not required to demonstrate any training to anyone. It's required to perform a service related to a disability, but the law does not allow any way of confirming that it actually does.
Under the ADA, there are two and only two reasons to eject a service dog, and "is a Great Dane" or "isn't a registered service animal" aren't among them.
"A person with a disability cannot be asked to remove his service animal from the premises unless: (1) the dog is out of control and the handler does not take effective action to control it or (2) the dog is not housebroken. When there is a legitimate reason to ask that a service animal be removed, staff must offer the person with the disability the opportunity to obtain goods or services without the animal’s presence."
As for those grocery store signs, here's what the ADA website tells businesses: "When it is not obvious what service an animal provides, only limited inquiries are allowed. Staff may ask two questions: (1) is the dog a service animal required because of a disability, and (2) what work or task has the dog been trained to perform. Staff cannot ask about the person’s disability,
require medical documentation, require a special identification card or training documentation for the dog, or ask that the dog demonstrate its ability to perform the work or task." Any store saying animals must be "registered" is making a mistake, and potentially a very expensive one.