Does turning the vehicle off and on damage it ?
No, that is what it is designed to do.
It does cause wear on components like your starter and battery and the more you wear those the faster they will need to be replaced. That’s why the newer vehicles that shut down at every stop go through those components so quick. Because it is considered normal wear the dealer warranty rarely covers unless they fail shortly after a new vehicle is purchased. It’s definitely a poor design that has only been implemented by auto makers to meet emissions standards at the cheapest cost to them. The customers are going to be the ones to pay by replacing starting components 5 times faster than they normally would if the car didn’t turn off at every stop.