Animals do always have to be under control of the owner though, so if one is not under control that is an acceptable reason to reject them.
And I believe the latest guidance is that basically only dogs fall under the service animal category, so random other animals people use for emotional support would not qualify either. The exception seems to be miniature horses for specific limited cases.
https://www.ada.gov/regs2010/service_animal_qa.html https://www.ada.gov/service_animals_2010.htm
There are legitimate service animals for a wide variety of real conditions that a lay person could not be expected to consider. Dogs that can detect incoming seizures or problematic psychological states, and they could look very similar to an 'emotional support animal' but actually be a legitimate service animal.
Also having personally seen how poorly trained most peoples' dogs are I would be very uncomfortable having a random stranger's dog near me. A lot of the dogs I encounter on the street are neurotic wrecks incapable of passing without barking, leash pulling, or lunging.
If they were misbehaving, that's reason for them to be rejected, training or not. If not, legally allergies and fear of dogs are not valid reasons for denying access or refusing service to people using service animals. Though Uber should certainly attempt to pair the rider with a different driver first, if they have registered that allergy / fear ahead of time.
This should change. Especially on the allergy front. Forcing someone to suffer an allergic reaction because a customer has a service dog is immoral.
But, unless they are the only one available or something, there is nothing in the law saying that specific employee has to keep providing service. The company should make a best effort to remove that employee from contact, without impacting the service of the customer.
I'll let you decide who has the better case.
There's a distinction between trained service animals and emotional support animals. Both are support animals but the latter are largely untrained which is why airlines have moved to ban them. I have no issue whatsoever with trained service animals!
I think there's a conversation to be had about dogs as vicious in nature but I have been laughed out of the room when I mention this to an in-person audience. But many USPS/UPS/FedEx and now Amazon delivery people are aware of this: untrained dogs are a danger