This Audi A4 Quattro is unique, and, as Akio Ogino states, has the special AWD feature, that allows them to take the old road in the first place. Although the car from the movie Spirited Away is grey, this color is almost impossible to find in real life. It is said that the engine noises from a real Audi A4 1.8 were used in the making of the film.
The car is a symbolism that the family has a western-oriented sense of consumption. This mentality is later on seen, when the parents eating at the Food Stall.