Uber and Lyft allow their drivers to accept or refuse passengers with non-service animals including cats. Both companies encourage riders to contact drivers prior to pick-up to learn about their policy on cats.
Uber: Uber allows service dogs to ride with their owners. Each driver-partner is allowed to decide whether or not to drive passengers with non-service dogs. Lyft: Lyft drivers allow service dogs to ride with passengers but are given latitude on whether to accept passengers with non-service dogs.
Uber and Lyft are required by the federal Americans with Disability Act to serve passengers with service animals. Both companies highlight compliance on their websites.