Probably not. If that were the case, then the LED lights would go off first. Ever look at the heat sinks on those things? Lotsa amperage. Ditto the AC and cabin fans. and the MINI display screen. All much higher current devices than a stereo system, even cranked.
I suspect it is probably just to keep parents from getting blasted by the kids' volume settings when they start the car, or something. The engineers surely understand that long exposure to high volume acclimates your ears. Remove the sound, and the ears become acclimated to the relative quiet. If you return the sound to the previous level, it will feel louder than it felt before. Smoothing out the user experience, IMHO.