I can bet that in most big European cities, relying on public transportation + renting a car whenever you need is cheaper than the maintenance costs of car ownership (parking + insurance + gas).
Making the car rental more efficient on a global scale will reduce the need and the utilitarian value of car ownership. There will be of course those that will want to have their own car, but that will be more and more considered a luxury item or status symbol than a need for anyone living in urban areas.
That's already the case though. Having self driving cars doesn't really change the equation on that.
I could live carless in Boston when I was living in Cambridge or downtown Boston. I could not make that if I wanted to live in Medford, even though that is a bare 10-minute drive from Cambridge. A fleet of self-driving cars could be spread around more efficiently in metro areas and suburbs.
I'm not a fan of most of his public transit projects, however there is nothing about them that dictates who owns the cars. It isn't likely that Tesla would block a Tesla owned Tesla vehicle from using those tunnels in favor of a privately owned Tesla vehicle.
>I find this idea that people would be willing to share a car with the whole world pretty naive, majority of car owners are likely to want to keep their cars for themselves.
Will everyone participate? Obviously not. However, the Ubers, Lyfts, ZipCars, Turos, and even the Enterprises and Hertz of the world show that many people would happily share a car given the right circumstances. There doesn't need to be 100% participation to reduce the number of cars in existence.