Why the ground effect has such a different outcome for wings and cars?
It actually has exactly the same outcome! Well, in reverse.

Wings generate lift by forcing air down (one way or another, let's not go into details). When the wings are close to the ground, the air that is forced down can't get out of the way, resulting in a higher pressure.

Cars generate downforce by forcing air up. When air close to the ground is forced up, it can't be replaced by 'new' air easily. This results in a low pressure, sucking the car down.

The ground effect is thus exactly the same in either case: air that is either pushed out or pulled in, cannot be replenished quickly enough due to the ground being in the way. The difference in outcome is because a wing is attached at a positive angle w.r.t the ground, whereas a car is shaped like a very thick wing at a negative angle.

A comparison could be a hovercraft on the one hand, which is lifted off the ground almost exclusively by ground effect (if you were to drop a hovercraft from a plane, it wouldn't matter much whether it was powered up or not). On the other hand, a vacuum cleaner, which is sucked down to the floor exclusively by ground effect (indeed, Feynman's sprinkler experiment will tell you it is not sucked down because of the upwards acceleration of air - only the 'engine' part of your vacuum is forced down by momentum exchange. Test this by holding the vacuum in the air - you won't feel it 'pull' at all).