I'm really not sure that this is the case at all. Taxis already exist in basically every city in the US, as do vanpools, bus services and heavily subsidized public transportation, but we still have significant rates of car ownership. Self-driving cars do improve on the "human-driven-car" model because they have faster reaction times and the like, but the aspect of self-driving cars where you don't have to be in control of the vehicle isn't anything new.

It's certainly possible that cost was the major driver of these sorts of things, but my guess is that the convenience of having your own car ready at all times will remain paramount. It's also plausible to me that airbnb-like services renting out self-driving cars will be common, which would simultaneously alleviate the need to own a car and reduce the cost of car ownership.

I imagine the switch to primarily self-driving car services will have a greater impact in cities, where the density will be high enough that the latency between wanting a car and getting one will be smallest. If a car has to drive for 20 minutes just to pick you up and take you to the store to get some milk, you'll probably just want your own car.