Is it permissible for the car owner to collect financial contributions from passengers (for fuel and maintenance/repair costs) in exchange for transporting them to and from work and home, knowing that they have no ownership in the car other than the right to be transported?

It appears from the question that the car owner stipulated that the passengers share the car's expenses, which is his right, but on condition that the compensation is known and agreed upon, because a lease is not valid unless the rent is known. If the rent is unknown, the lease is invalid, and a fair wage becomes due upon the utilization of the service.