We can color the milk by either adding something, or taking something away.
The white color of milk is caused by particles and droplets in the milk that reflect (scatter) the light that hits them. If we remove the droplets of fat from whole milk, it takes on the slightly bluish tint of nonfat milk.
If we add brown powdered cocoa, the cocoa particles absorb a good deal of the light that hits them, but they reflect enough red and yellow to appear brown. The milk adds white reflections, so the result is a lighter shade of brown than the original cocoa powder had.
We can make pink milk by adding red food coloring. The red dye absorbs green and blue light. The milk adds white, giving us pink.
Because of the effect the droplets of fat have on the light, it is hard to get deep dark colors when adding dyes to milk. Even adding India ink only makes a gray color.