Does using the auto engine start/stop feature of a car actually damage the engine, and does it actually save gas?

I recently updated my car from a 2007 Subaru Forester to a 2023 Forester. The new model now has a feature in which I can turn off the engine by pressing down the brakes when at a stand-still, as opposed to letting it idle. The engine automatically turns back on when I let go of the brakes.

Does turning the engine off and back on damage the engine (over time) if I do it every time I hit a red light? Do the brakes get worn out? Does the battery get worn out?

Some energy from the battery is used to turn the engine back on. Am I actually saving fuel if I turn the engine off at every red light, compared to the amount of energy needed to restart the engine? Does the amount of time the engine is turned off make a difference (eg turning the engine off when the red light will turn green in 5 seconds, vs turning the engine off at a new red light)?