California law does not exempt taxis from the child safety seats requirements that apply to regular passenger vehicles.
As to the logistics of using a car seat in cabs and ride-share vehicles, I haven't a clue.
Edited to add: Apparently, Uber does have a car seat option for an extra charge (
Lyft has car seats available only in NYC).
Airport Taxi Cab SFO SJC OAK also offers car seat taxi service.
P.S. You may get a better response in the
Travel with Children forum.