Can an uninsured driver be legally held responsible for damages to other cars even if the car he is driving is insured?

Yes, Both the permissive use driver and the vehicle owner(s) are financially liable jointly and severally for 100% all damages and injuries sustained in an "At Fault" accident. Both can be sued.

If you were driving and at fault. Then you are the person responsible for causing the accident and you are financially liable.

If the owner gave you permission to drive the vehicle, although he's not at fault, he is also financially liable because it's his car and he's the one who allowed you to drive the car.

If you are not a covered driver under the terms and definitions of the owners policy then the insurance company will not pay for the accident.

We should never borrow or loan out a vehicle without first checking if the driver will be covered under the policy.