Other disability "credentials" such as parking placards are issued by the states. The ADA has minimum standards for disability but states are allowed to be more lenient, and some are. California, where I live, requires a doctor's (or certain other medical professionals) signature one time only, for the first placard. Now and then someone says the cure for placard abuse is to require re-certification on a repeating basis. Those of us who have placards argue, why should we be penalized? We believe first the state should try enforcing what they've got. We believe burdening people with legitimate disabilities should be a last option, not the first one.
The same issues apply to fake service dogs. Professionally trained dogs are expensive and there are not enough trainers to satisfy demand, that is why people need to be able to train their own. Any standard test would not be appropriate for all dogs because they are trained for so many different disabilities. About the only things that are required of all service dogs are basic good citizenship such as no aggressive behavior and house-broken. After that, some guide, some pull wheelchairs and fetch out of reach items, some react to low blood sugar, some react to impending seizures, etc. Licensing or testing service dogs just doesn't make sense because of the impossibility of standardization.
The idea of certifying that the user has a disability makes some sense but figuring what bureaucracy is going to handle it is not so simple. Even defining disability in a way that makes it easy to make decisions is problematic. For parking, the issue is, does this person need special parking for some reason, either close-in or a wide spot. Many people are in the protected ADA class but do not qualify for a placard, at least in California. I can't imagine what the rules would be for a service dog. Just what boxes are going to be offered to the doctor to check? Remember medical professionals are not lawyers and have no idea of the fine points of, "What is a qualifying disability?" "What is a service animal?" and "What specific task would a service animal perform that would reduce or eliminate the effect of the disability this person has?"
Asking a vet to certify what training an animal has had is also problematic. First, my my vet has no business knowing my medical history. In addition, my vet does not have security in place to protect my medical info. (HIPPA does not apply to my pets!) In addition, I don't know about other vets, but mine wouldn't certify any training unless he knew the trainer personally.
Any paperwork that does not have to go through a centralized clearinghouse and that is accessible to anyone who has to enforce these rules (conductor, grocer, restaurant manager, etc) is just going to encourage a fraud-based economy, like has developed for parking placards. In my opinion, one quality of a "good" law is being easily enforceable, and certifying either service animals or their human users doesn't pass yet.
So we muddle along with a bunch of cheating. I think Amtrak has it right, punish the behavior. It is a real shame that we live in a culture where cheating of all sorts is acceptable, where some people make a risk/benefit calculation and decide it is okay to cheat because they probably won't get caught and the penalty is tolerable.