Businesses and what not most certainly can ask if your animal is a service animal, they just cant insist you provide proof.
The two big questions that are permitted are "Is the dog a service animal required because of a disability" and "What task has the animal been trained to perform". Also worth noting that the ADA is subject to "reasonable accommodation" clauses, which state that while the business must make reasonable attempts to accommodate a service animal, they are not required to do so if it would cause the business undue hardship or threaten the safety of others present (for instance, service dogs may or may not be allowed on ambulances due to the restricted space).
Also also, to the best of my knowledge dogs are the only animals that can be trained as true service animals. While anything can be an "emotional support animal", those are not covered by the ADA.
Noop_Noop posted...
The only 2 exceptions to this that I am aware of are airports and federal centers
Religious centres (churches, mosques, etc.) are also exempt, as are any businesses that can reasonably claim that the presence of an animal would fundamentally disrupt or alter their business (parts of zoos, for instance).
Ponies, cats and pigs can all be trained and certified service animals.