A lot of people want to paint one size fits all judgment on ESA and indeed some dogs in general. One poster mentioned Europe. I just returned from a 35 day trip by RV across Europe. I encountered well behaved dogs, not ESA or service animals at several restaurants. They were welcomed and tolerated and behaved well laying down on the floor and minding their own business. In fact in a very busy restaurant down in the Bodensee in Germany, the wait staff brought a little beagle a water bowl inside the restaurant before the patrons. NO ONE SEEMED TO CARE. It was wonderful. I asked the owner about this and he said, "most dogs are behaved better than the children that come in." Harsh, but probably true.
Not all dogs cause allergies, my Biewer Terrier gets regularly groomed and does not shed. On a flight from MSP-TUS I had my dog in coach on a paid fee. I was in a window seat and he was asleep in his carrier. The woman on the aisle seat was griping during the flight about being allergic to dogs. Towards the end of the flight, I lifted his carrier and put it on the tray table and let him poke his head out (I know, against the rule, but he is 6 pounds and wasn't going anywhere). She was shocked at first but then had to admit she did not even know he was there. I have flown first class on Delta too with him and unless they physically saw him, they did not even know he was there.
Lastly, I have a priest friend who has battled a couple people who claim they are allergic to incense at funerals and other Masses. One time, he brought the censor out and started waiving it around. A couple people in the front row started to cough and make exaggerated motions. He then paused, and lifted the lid too show it was empty. Was it appropriate? Maybe not, but point made. Some people are genuinely allergic and some people are emotionally allergic. We all need to be responsible for our emotional and physical health, and if we are deathly allergic, always carry the proper medicine.
The airline was in a no win situation, the complainer was wrong. I am also a T1 diabetic and could probably get a true service dog that detects low glucose. I am not bothering because there is a technological solution using a CGM. However, if my dog was trained to detect and alert, how is it not my right to travel with him? The end result is the same, a dog in the cabin. One paid, one didn't.