I find that having the A/C on with the heater helps defog the car faster in the winter. Is it possible that doing this can damage the car?
No way. In fact the system is actually designed that way.
The Air Conditioning (A/C) system on your car runs automatically when you are in the Defrost mode. Like you say that setup removes water vapor from the air preventing fogging inside the windshield. That’s considered a safety feature.
The system has internal shut off protections to starting the A/C compressor when its super cold outside. Those protections use refrigeration system pressure and/ or temp sensors for that purpose.
Note also that the heater system will totally overwhelm the A/C system. In fact many vehicles blend in warm air to perform automatic temperature control.
Not to worry. You will not damage your car.
Absolutely not. On the opposite, it's recommended to do it from time to time in the winter.
In fact, turning recirculation and hot air on at the same time together with AC is a way of "running-in" a compressor that has just been installed on a car after flushing the AC system of any old oil and filling the compressor up with new oil. Why? Because the additional heat load forces a thermal expansion valve wide open and the oil circulates back to the compressor (after having filled the other AC system components along its way) much faster. Or, in a cycling clutch orifice tube air conditioning system, it keeps the compressor pumping for a longer time by keeping the suction pressure up. Variable displacement compressor with an orifice tube as the expansion device? It keeps the compressor running in full stroke mode. See? It basically maximizes the refrigerant and oil flow through the system in a condition in which this flow would otherwise soon stop (to keep the evaporator from freezing up) one way or another, depending upon how the air conditioning system has been designed.
Hence, it helps keeping every seal in the system lubricated, preventing refrigerant leaks from springing up, and prolongs the AC system life.
I have an old car with a TXV as the expansion device and a suction pressure-controlled variable displacement compressor which recently received a new control valve, to keep it working like it's supposed to do. If i turn AC on in a cold winter day with fresh air set, the engine soon begins to act like the AC is turned off even if the compressor is still operational. If i then switch to hot air and recirculation, the compressor starts loading the engine up and slowing the car down like it normally would in the summer.