Once automated cars are in place, companies can simply become dispatching services for privately owned cars. Don't want to pay for parking in the city after your car drops you off at work? Let it drive around town and make money for you, then pick you up at the end of the day to drive you home. Then, it can pickup your neighbors as they leave the local bar at night, and it is ready for you, charged up in the morning.
The point is that if your/Uber's car is making money for you, it is a business, whether you call it a taxi or not.
And despite what libertarians here like to pretend, there is a difference between business and personal activities, and yes this involves Evil Government Regulation and even, oh the horror, taxes.
Amusingly, a vast fleet of non-personally-owned automated cars seems more like socialism than the US's traditional rugged individualism of private vehicle ownership, but hey ho, maybe something good will come out of Uber (once they're nationalised).