If you were to put both leads on negative and positive terminals on car battery while connected and have the meter on continuity (the one which beeps), what happens and can it fry/ruin the electronics in a car?
It's unlikely to do the car electronics any harm at all. The car battery can put out hundreds of amps at 12V if it needs to. A weedy little continuity meter will have little effect on it.
But it could well fry the continuity meter if it's poorly designed. A better designed one will just go into overload mode.