If I see a dog in a store where dogs aren't allowed, I'm going to assume that dog is a service dog.
If someone has a well behaved dog, I probably won't even notice them. By law, business owners/employees aren't supposed to ask more than a couple of questions:
When it is not obvious what service an animal provides, only limited inquiries are allowed. Staff may ask two questions: (1) is the dog a service animal required because of a disability, and (2) what work or task has the dog been trained to perform. Staff cannot ask about the person’s disability, require medical documentation, require a special identification card or training documentation for the dog, or ask that the dog demonstrate its ability to perform the work or task.
A badly trained service dog (or a faker with a badly trained pet) is probably going to be noticed more often (and more easily) than a well behaved pet.
I disagree with bringing pets into stores where they aren't allowed. It's something that I would never do.
However, I still stand by saying that I'd rather see a well behaved dog at a store rather than a badly behaved service dog. I'm not going to notice the well behaved dog (except to maybe think to myself that it's awesome and doing a great job), and most people are probably the same way.
I've seen some pretty bad service dogs (or fakers- who knows, I would never ask), and I think those are the ones who are doing the most disservice.