Then Amtrak pays off the damage to the vehicle just like any other vehicle, it's just that the rental company that owns the car will get the settlement rather than the driver. I see no issue with this, as long as the value of the damage is covered by the party responsible.
When the auto train derailed in 2002, I'm sure they figured out what to do with the damaged vehicles, and I'm sure that all of the employees still around from that time would help management to get it done right, as upper management seems to have changed more than once since that incident.
However, Amtrak may pay less than what the rental company wants in damage reimbursement from the renter. For example, if the (rented) car is a write-off due to a derailment, Amtrak (or its insurer/agent) might only reimburse to the book value of the vehicle while the rental company may insist that the renter or his/her insurance policy pay for a full "replacement" value. In these circumstances, an AT passenger with a rented car could get caught in a $$$ gap between 2 large corporations with little leverage to get one corporation or the other to cover the gap and have to cover it out of pocket.
Of course, if one is riding on an AT that has a derailment bad enough to damage the automobiles, the damage to the rented car may be way down one's list of immediate concerns.