Status symbols are inherently generated from scarcity. The point with a distributed autonomous system of driverless vehicles of any and all kinds (whether they be cars/buses/hyperloop/whatever) (also not arguing that a hyperloop is a near-term reality) is that we essentially increase safety, decrease cost, and improve efficiency for everyone.

Do I think that no one will own a car? No, I don't claim to know that.

Will there be a radical shift in our culture towards relying on autonomous self-driving cars? Undoubtedly, and this will start to change notion of ownership. This may not change the culture of car ownership immediately, but it probably will in the long-term.

At one point in the history of humanity, having a man-made source of light was a status symbol. Then we made light abundant (at least in the developed world). At one point, having clean water was a status symbol, now we have made water abundant (again, at least in the developed world). The same could conceivably happen to transportation. The mechanisms exist.