Actually, it’s rarely the starter that gets damaged. It’s more often the flywheel — which is the larger gear that the starter gear meshes with.
They’re both pretty tough, and accidentally starting the car when it’s already running once in a blue moon probably will not do any damage. If it’s done repeatedly, though, you can wear down or break a tooth on the flywheel.
And if you have a flat spot on the flywheel, where a tooth or two is missing, the starter gear will spin but won’t be able to turn the flywheel, which is what moves the pistons up and down and starts the car.
Even when that happens, by trying to start the car several times, you often can move the flywheel just enough to get it to the next tooth and allow the starter to catch.
But at some point, the car won’t start – or will start so unreliably that you can’t take it anymore. Then you have to replace the flywheel.
Replacing the flywheel usually involves removing the engine, the transmission or both. That’s why mechanics refer to that horrible grinding noise as “the sound of money.” So it’s best not to make a habit of restarting a running car.
By the way, many of the newer cars with “start/stop” buttons instead of keyed ignition switches won’t even let you restart the car if it’s already running. They’re smart enough to know when the engine is already running, so if you push the button a second time, the car assumes you changed your mind about going to work, and it shuts off the engine.