YES. While businesses are generally permitted only to ask whether the dog is a service animal required because of disability and what the animal has been trained to do, there are instances when more extensive proof can be required.
If you file a complaint about discrimination, proof of disability and proof of training will be required. If you appear in court and you claim to be disabled and claim your dog is a service dog as part of the case you are involved in, then you will have to provide proof that your claims are true. A court will not simply take your word for it.
If you are arrested for trespass for bringing an animal into a place where pets are not permitted, your affirmative defense may be that the animal in question is a service dog, but again, you will have to prove that is the case.
Proof may include:
Medical records from any medical providers treating you for your disability or for aspects of your disability.
SD certification from a recognized/accredited program.
Training logs if owner-trained.
Independent evaluation of your dog's training by a qualified trainer.
Certificates attesting to training and temperament, such a training class completion certificates, an obedience title or certificate, a CGC certificate, etc.
Video demonstrations of the dog's training.
In person demonstrations of the dog's training.