No, you are not taxed on the proceeds from an insurance property settlement for an auto accident.  The settlement is being used to replace the damaged property, it is not considered a financial gain.

The fact the car was fully paid off does not make it taxable. If your car was totaled, the insurance appraiser pays you for FMV of your car before the accident.  If the insurance company paid you more than FMV for your car, you would have a taxable financial gain but they don't pay more than FMV.