That's actually backwards. Once you have self driving cars, it will be cheaper for most people (starting in urban, then sub urban areas, but ultimately everywhere) to just pay for taxi service rather than make a major investment in an asset with about a 5% utilization rate. And in that case the companies with the best financial engineering will succeed. Insurance companies will prefer to insure fleets rather than continue to do actuarial analysis on individual drivers, especially when you can continuously "train" (i.e. upgrade) your whole fleet. Basic safety will restrict driving to a small number of people who absolutely need it and are willing to take the extra training and liability. I will need to do this for off-roading but boy I can't wait to stop road driving.

Uber has a customer base and minimal physical plant, which is a huge advantage for them. On the other side, car manufacturers have enormous balance sheets and can be vertically integrated travel providers -- and have a huge incentive not to simply become manufacturers for fleet operators, which will put them in the wrong part of the value chain (basically the equivalent of "dumb pipe" telecom carriers).

Most of the car manufacturers will go out of business. My guess the survivors 20 years from now will be three from the set of Ford, Daimler, VW, Toyota and maybe Renault/Nissan and GM. Almost certainly Daimler will be on that list, not sure about which of the others. And the rest are already walking dead.

By the way cars will become quite boring commodity products (like planes) because the users and the buyers will be different.