Therapy dogs and service dogs are not the same.
Service dogs are trained to help with a disability or do a specific task. Guide dogs, PTSD assistance dogs, medical alert dogs etc They are well trained and unobtrusive. You will not see service dogs pulling on leads and making a nuisance of themselves. They will always have a jacket, harness etc stating that they are a service dog and any misbehaviour would be swiftly corrected.
Therapy dogs are just a dog who is a comfort to the owner. They are not trained to do anything specific, sometimes not trained at all! They may wear a jacket with Therapy Dog on it but these are sold on Amazon and hold no weight.
You cannot deny entry to service dogs. Legally they are permitted to enter.
Therapy dogs have no rights to go anywhere any other dog can't. You are allowed to refuse access. Whatever the owner says, it's bullshit. Some places choose to allow therapy animals but it is a courtesy, not an entitlement.
As for the specific example you gave: bollocks. The dog would have been raised to wear the jacket from a young age so would be used to it. If he could not tolerate it, he wouldn't have made the grade as a service animal. She was taking the piss.