I'm pretty sure the bit where it says there's chainmail under it is just describing the armor, not saying you combine the two types of armor. It's just like how Full Plate is described as being worn over a thick layer padding. This doesn't mean you get to add the bonuses from Full Plate and Padded together, it's just telling you how the armor looks.

Besides, even if you did wear two types of armor at once, armor bonuses don't stack. Armor check penalties might, though.