There will be cheap fleets and more expensive fleets, just as there is UberX, UberBlack and UberSelect. You could also imagine a BMW fleet that offers specific models only if you are an established frequent customer, so people can show off that they didn't just upgrade for this occasion but always travel in style.

Of course there are still plenty of edge cases where individual car ownership is nessesary, whether that's because too few cars of that type would be booked to make a fleet viable (Rolls-Royce) or because you are in a very rural area.

What you are describing is an automated taxi.

Rich people who want to call a chauffeured luxury car can do so in most major cities. Many of them do use such a service, but many of them also own expensive cars to show off their wealth.

I would expect that their status would come from being able to hail more expensive cars. They would spend extra to have an automated Porsche, BMW, etc. pick them up.