It is your right to take an Uber, but I don't see why it should be a right for your dog to take an Uber. The Uber service is based around private individuals using their own cars to make some money and many are just doing it casually. I have a dog and I love him dearly, but he doesn't go in my car. He goes in our second car because that is suitable. My car however is suitable to carry people and I could use it for Uber if I so desired. It has a leather interior and I would not allow any dog into it.
I have no doubt your dog would be very well behaved. I also have no doubt it would not relieve itself in the car. However, your dog's nails could damage my leather seats and it would leave hair which would mean my wife could never come in the car as she is highly allergic to almost all breeds of dogs.
So whilst I understand your have rights and I do empathise with your situation, you should also understand that the Uber service is not a taxi service per se.