Aside from fire, theft and personal injury, insurance covers physical damage resulting from a collision. It does not cover damage caused by misuse of the car, such as popping tyres on kerbs or knackering the clutch or gearbox out of clumsiness.
The instructor is trying - in his own unique way - to get you to understand car sympathy, most probably because you are being rough and careless with his property, which would cost him a pretty penny if damaged. If you saw somebody throwing a ball close to one of your windows, you'd be somewhat twitchy too, and would probably warn the person throwing that they "wouldn't want to pay for a new window".
If you did damage some part of his car despite numerous warnings not to, he'd be perfectly within his right to ask you to contribute to the repair. Some instructors may even write this into their contracts with you.