I see little reason to suppose that that will be the end game. Those for whom the total cost of ownership of a car are lower than the cost of taxis will continue to own a car.
What fraction of people that currently own cars will that be? Probably depends on where you live. Taxi will probably never be the popular choice in, say, South Dakota. The economics just don't work out.
The cost of autonomous cars, especially in terms of maintenance, is likely to be higher. And autonomous taxis will be cheaper to operate because drivers are a very expensive part of them. The number of people for whom owning a car makes sense will definitely go down when prices on cars rise and prices on taxis and public transportation drop.
This is ignoring the fact that having an unproductive self driving car sitting in your garage becomes a significant opportunity cost itself.
What fraction of people that currently own cars will that be? Probably depends on where you live. Taxi will probably never be the popular choice in, say, South Dakota. The economics just don't work out.