Yes.

The law requires every driver has liability insurance.  Which means the insurance that pays damages to others caused by the driver. 

When his mother put his name on her insurance, that insurance probably covers his liability (check with insurance first, cuz I don't see his paperwork), no matter what car he's driving.  For example, when you rent a rental car, you don't buy any insurance because you are coverred by your car insurance to the extent of damages you caused to others. 

But your liability doesn't cover the damage you sustained yourself (e.g. damage to rental car or the newly bought car), that's why the rental car company sells you extra insurance to insure the rental car.  If you declined that extra insurance, you'are liable to rental car, but you are not liable to damages you caused to other cars.