It has been explained to me that cars are an extension of ones personality, a social signal, and a hobby. So for some apparently what kind one rides in is significant.
Much like buying a museum ticket to see a famous painting isn't comparable to owning the same painting.
I haven't seen Uber or Lyft effectively go down the route of having their service be a part of a user's identity. But Waymo could make an effort, if they wanted to: they use a distinctive car model, and they could offer incentives to make carpooling with Waymo cheaper (to try and make a user 'known' in their social circle for using the service).
Some limos have poor or blocked views of the outside which can trigger motion sickness. I also wonder about their side impact protection and how much testing that gets.