No, while the work the engine does would be reduced by a tailwind, it would not be reduced to be equivalent to the relative speed travel.
Work $\ne$ Force
The work that the engine must exert to maintain speed is equal to the drag times the velocity. So let's look at the ideal situation where the only drag on the car was due to the wind. The drag on a car going 60 mph with a 15 mph tail wind would indeed be equivalent to the drag on the same car going 45 mph in still air. However, the work that the faster car would have to do to overcome this drag would still be $\frac43$ as much as the slower moving car.
Interestingly, this is not the case with the airplane. This is because while the car pushes on the ground, the airplane pushes on the air. When there is a tail wind, the airplane doesn't have to use as much energy to impart the same amount of momentum on the wind.
The car engine must overcome other resistance forces besides air resistance. The rolling friction of the tires, the engine and drive train friction, and others will contribute to the amount of work the car must do to maintain speed. Many of these sources of friction increase with wheel speed, and thus velocity of the car, so it would not only take more work and power to maintain the 60 mph, but also more force.