Yes, boybi ... some new cars have electronic control for the A/C and will disable the compressor when engine is not running.

However, every manufacturer recommends that A/C should be turned off when not in use.

Other than being an additional electrical load for the battery while the engine is not running, and as additional mechanical load when the car is started, I really don't see why this practice could possibly cause any damage to the car. Sure, the car may have difficulty starting (which justifies manufacturer's advice), but there is nothing that would tend to break. You may have consumed more electrical energy and the starter operated with higher current to start the car, but both battery and starter are so durable, the additional load is peanuts.

I guess the recommendation was meant to avoid ... peanuts?

The lolo is correct. Matibay ang oto niya kagaya ng oto natin.