To add, back in the old days when I still went into the heart of my local metropolis, Uber and Lyft style carpooling had the opposite effect of "reducing the number of cars". The streets are littered with pee bottles and packed with desperate drivers looking to score a quick buck, making u-turns across traffic while looking at their phones.
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.
Personally, I think a lot of people would take that trade. I would.
Yes, there are people who like lugging everything but the kitchen sink with them around. For a lot of people in the world though "things you always want to have with you" fits in a handbag or backpack.
It's also about enabling Uber-style intelligent carpooling so that you can save energy while still maintaining similar times to driving, especially in places with poor or non-existent public transportation.
It's also about reducing the number of cars in the world. You probably use your car for less than 5% of your life. Most of the time it just sits there taking up space. Why do you need to own it? Imagine a world where you didn't need to own, maintain, or park. We can reclaim parking lots and plant trees, build housing, or other things. We can also reclaim a lot of our time spent in traffic jams or looking for parking and do something more useful with it.
It's also about enabling people who are unfit to drive to have a better quality of life. What if you've had a little to drink? What if you're down with a fever and a headache and can't concentrate on the road? What if you're sleepy? What if you are prone to seizures? What if you're blind, deaf, have poor motor skills? What if you're a visitor to an area without a public transportation system and not confident renting a car and driving?