Summary: As at the date of this post the answer is No. Except for Sydney and Brisbane in Australia or except for one service animal anywhere plus protection for the car.
It seems that until very recently (in Australia) companion animals except for service animals were universally not allowed in Uber cars. But there is new service called Uber Pet which is being trialled in Sydney and Brisbane. The service costs an extra $7 per ride (not cheap).
Dogs and cats must be on a leash and other animals must be in a carrier. To reduce mess in the vehicle a blanket or some kind of protection should be used. A carrier for the service animal is an alternative. There may be a charge if mess is created. You can bring one companion animal on an Uber trip.
I’d expect cat guardians to take their cat to the vet in a carrier! So what’s the answer to the question? The answer is No. Except for Sydney and Brisbane in Australia or except for one service animal.
As mentioned service animals are allowed because there is a legal obligaton to do so, but you should contact the driver to let them know. I’d expect the legal obligation for taxis to take service animals to be a universal rule.
But service animals are not allowed on uberPOOL rides. This is when people are travelling in the same direction to share the costs.
P.S. Lebanon has a pet taxi service and it is cheaper and more relaxed than Uber Pet!
How to prepare for travel with your pet to any EU country whether the UK leaves the EU with or without a deal