Waymo says they are a premium service like Uber Black because they have nice cars (Jaguar I-Pace), plus the novelty and safety of being driverless. They’re not trying to be competitive with Uber X or shared rides for in their current form.
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.