But now let's say this car with the person inside fell off a cliff. If he let go of a ball in his hand, the ball will not go to the back of the car.
First of all, there is not equivalence between a car accelerating at $9.8m/s^2$ and a car falling down the cliff.
The equivalence is for a car accelerating and a car is at rest on the surface of the Earth (of course the latter should be oriented upwards in order to get the same directions), due to the reaction force of the car itself to the person or the surface, respectively.
A freely falling car does not feel a net force, it would be like it is at rest in the space far far away from any kind of gravitational potential.
If the car was moving at a constant speed, only then it would be just like a free falling car. And the ball would not go backward or something inside the car in both cases here.
On the other hand, inside a car at rest on the surface of the Earth (assuming it is oriented upwards somehow) the ball will simply fall towards the center of the Earth (which is the back of the car in that weird orientation) just like the ball inside the car that is accelerating at 9.8m/$s^2$.
The equivalence principle is a part of the Newtonian gravity as well as Einsteinian. Because it is first postulated by Galileo. Newtonian mechanics sees the principle in terms of force as expected.
Indeed, in Newtonian picture, you only feel the presence of the gravity because you are exposed to the reaction force by the ground you stand or the chair you sit. Otherwise you would freely fall just like being at rest on interstellar space.
So does the person in the accelerated car, that is, the car is accelerate the seat, the seat resist due to its inertia so it encounters a reaction force by the car, and the driver does the same because s/he is resisting to the seat's acceleration and encounters a reaction force and so on. Just like an object on the surface of the Earth tries to fall but is stopped by the ground as a reaction.