Milk is mostly water, but it has some milk fat.
This situation is referred to as a colloidal suspension and it is rather common. When the food coloring is dropped into the milk the tiny balls of fat in the colloidal suspension prevent it from spreading throughout the container and they stay, more or less, in one place. As the detergent dissolves some fat molecule balls others move in to replace those interacting with the detergent. They carry the food color molecules with them and the whole system seems to churn with amazing colors. Only one drop of detergent is needed.