I do. I want to store stuff in it (my assorted gear I want nearby wherever I go but won't be lugging around every time I leave the vehicle), I need specific features (say, child seats), I want to customize it (be it bumper stickers or off-roading bumpers), I want to know that there WILL be a suitable vehicle ready when I need transport ("whadayamean there are no vehicles available with two child seats? my girl needs a doctor NOW but an ambulance is super-expensive overkill!" or "it's 1AM and I have to wait 15 minutes for one to arrive? with my own car I could be home in 15 minutes."), and I don't want to deal with consequences of strangers driving it before me (everything from annoying seat settings to vandalism & bodily fluids).
Sounds like you're thinking of a self-driving taxi, overlooking the many reasons why people don't want taxis (not deeply relying on others being a huge one for many of us).
I don't think anyone's trying to pass laws banning owning one's own vehicle once self-driving cars become ubiquitous. I imagine you'll always be able to own your own car, and store whatever in it.
(I also suspect your insurance will start to climb through the roof once self-driving cars do become ubiquitous, because computers will be able to be far better drivers than people, once the software's in good working condition.)
If 90% of the cars are taxis, there won't be much of an issue with availability. The cost difference should be huge too.
I do like driving around with loads of crap in my car though, I think some kind of standard secure box storage system would be helpful, a wheeled box you can lock up, like a bike rack, standard size so you can easily fit 4 of them in the taxis
Sounds like you're thinking of a self-driving taxi, overlooking the many reasons why people don't want taxis (not deeply relying on others being a huge one for many of us).