We just returned from a 5 state trip and stayed in a different hotel each night. We were asked to show Bella's id more than once and one hotel would not let us stay until we supplied the dogs service dog id for them to make a copy. Bella always has on her service dog vest but we were asked many many times in all 5 states for a service dog id.
This is not an uncommon experience, but it is still illegal. No one in any venue at any time can require proof the animal is a service dog.
I am not sure about hotels, but I do know that emotional support animals, which are more than pets but not trained to provide a specific service, DO require documentation before they are allowed into apartments which do not normally allow animals.
Service dogs are not required to have a vest, identification, certification, documentation, or anything else like it. As stated before, the venue can ask if the animal is a service dog, and what service it provides. Nothing more. I do believe that the animal is also required to be on a leash while in public, but that may vary state to state.
A venue may ask a person to remove their service animal if the animal is disruptive or dangerous - barking, growling, not on a leash and wandering, licking, being allowed on tables in restaurants, etc. The person must be allowed back in, but a disruptive or dangerous animal can be asked to leave. This is the only time a service animal is not allowed into a public place.
There is no such thing as a "universal certification" for service dogs, though different programs do indeed have their own certification.
It is very common to be asked to provide "proof" your service dog is "real" but it is illegal to ask for or require documentation.