>> Why do you need to own it?

Because our primitive brains instinctively want to own things. We can share with family, and with good friends, but not with strangers.

Another thing: car is used not only to drive, but also as a storage space for things that you want to have always with you: a bottle of water, duffel bag with gym clothes, umbrella, etc. With rented car pools you'd have to carry all that with you.

Also, society where no one owns a car or an apartment, where everything is rented, "pooled", would be a totalitarian dystopia - try to misbehave, post a negative comment about government on Facebook and you're banned from a car sharing service as a punishment. Seizing your property is of course also possible, but much harder.