I doubt it. Real people, especially parents with children, like to be able to store stuff in their cars. They don't want to deal with waiting, or dirty cars used by random other people.
Of course, the service is only a real alternative if a car is there within 5 minutes for you. People wouldn't tolerate more waiting.
It will be a gradual process and some people will still prefer to own a car, but in the long run owning a car could be a luxury if the driverless taxi really matures and it becomes much cheaper.