I think you need additional assumptions to draw this conclusion.
Imagine this: the cars on the road are originally unpainted, but just before you observe them a demon decides what color they will become: red or some other color (of which there could be many). The demon does not have a clock (to ensure that whatever he does is independent of time), but he does have a fair coin. Ordinarily he flips the coin to determine whether the car will be red or some other color. However, his can of red paint can color ten cars and needs to be used as quickly as possible. So, if the coin decides red, the demon just goes ahead and paints the next ten cars red. Once the paint is exhausted, he goes back to flipping the coin. Other colors are painted in the same fashion.
Obviously the car count per unit time is unaffected by this procedure, but I hope it's just as obvious that the count of red cars per unit time is not a Poisson process, because it will be overdispersed: there will be too many high counts, due to the temporal clumping of red cars, and too many low counts, due to the temporal clumping of non-red cars.
It could be that this scenario violates your sense of "the chance of red being independent of the color of the other cars," but it's difficult to know exactly what this statement means, since it's open to various interpretations. In the present case, the probability that the next car to be observed is red, given that the previous car was non-red ("the other" colors), is independent of the color of the previous car.