I'm not talking about cars as status symbols, but as factors of identity. Teenagers want cars because it is a rite of passage. Half my coworkers want trucks because they are 'outdoors' people. And some do use it as a monetary status symbol. Some people are just car people who have strong opinions about the car they drive because it is part of their self-expression. This is a pretty big thing in American culture.
Those people might use a vehicle subscription, because the cost of private rides would still be lower than the costs of private vehicle ownership (depreciation, maintenance, insurance). Many people would pay for the privilege of a private ride (like they do with Uber/Lyft now).
If car subscriptions services succeed and are nearly ubiquitous, the personal-freedom symbolism of car ownership will degrade.
Then, if people want to own a car as a status symbol, they will probably be better served by a more expensive one anyway, the same way people buy other status symbols (rose-gold iPhones, etc). Ironically, most of those status-symbol cars will still have an autonomous mode.