Eliminating car ownership makes a lot of sense in any urban area, but I bet people will continue to hang on in rural and some suburban areas.
In rural areas, an autonomous vehicle would need to travel long distances to pick up and drop off folks. It could be prohibitively expensive or inconvenient. People also use vehicles for other things in these areas, both work and play, which do not fit into the taxi model.
In suburbia, the taxi idea could work, but there might be advantages to owning your own vehicle. Maybe a family will own one autonomous vehicle, just one, but it would be in constant use throughout the day: mom goes to work, dad picks up donuts for coworkers, son goes to school, daughter goes to soccer practice, son goes to track practice, dad goes to get groceries, mom goes to get dinner, son goes to friend's house, daughter goes to football game, mom goes out to gym, etc. You'd have your one family car zipping back and forth between all these events in the family, picking people up, dropping them off, going off to do something else, and coming back again. Having an autonomous car in the family could make sense depending upon the costs.