Service dogs(or rarely horses) can generally go anywhere the owner can go including areas pets are prohibited. There is no formal verification, but the owner of the property or their employee can ask if this is a service dog that is required because of a disability, and what tasks it is trained to perform. If it is a cat or other animal, or not trained to perform a task, just to provide support it is not a service dog and they can be refused entry and trespassed.
Registration of service dogs would disenfranchise the disabled. About twice as many don't have photo ID as the general public, many do not own a car or have access to reliable transportation, even relatively small costs can be prohibitive
Service animals have to be licensed, trained to perform a task or duty, and be easily identifiable. Misrepresenting an untrained and unlicensed animal as a service animal is fraud and illegal.
I do think it would be better off for everyone if there was a license though. I've known people with legitimate service dogs who have had that opinion even. They wish they had a document to prove their legitimacy, as they are constantly asked for a document that doesn't exist.
The correct solution is for businesses to grow a backbone and do that for all animals all the time until people gaming the system with fake ESAs give up.
In the US this is not true by the ADA as licensing is a pretty prohibitive cost for many people with disabilities.
I personally don't see the argument that a separate license is needed or helpful. It's an extra cost and step that's entirely unnecessary. If you want or need a service animal you obviously have to have some way to train it and care for it, but a license is extra. Also if it is ever lost or anything you could be turned away from any public place until it's resolved. I just see it causing more issues for people than it tends to solve.
The thing that bothered me more than anything about this is that you aren't allowed to ask for any type of verification, so people can just walk in with them anywhere. Never made sense to me.
At least to park in a handicap space you need a hang tag or a special license plate. Why not something simple like a special collar tag or card in your wallet?