The differences we create in terms of race are purely man-made. Do we consider brown horses as a different "race" than spotted horses? The horses certainly see no difference.
With horses, dogs, cats, etc. we call them breeds because their offspring maintain those characteristics.
So, we actually do have various "races" of horses, some highly prized, such as the Arabian and Morgan.
The brown of brown horses is HAIR color, not skin color. And, no, we do not consider blonde caucasians to be a different race from brunette caucasians.