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.
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.