It seems odd that horse armor stacks, when regular armor does not. This is not technically a bug, but it is an inconsistancy.
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Horse armor doesn't have durability / take damage, while player armor does, which makes stacking difficult codewise. This is also probably for balance reasons: carrying more armor provides an advantage, because you can swap it out when it breaks. Carrying more horse armor just reduces tedium, allowing you to armor more horses with one load. Since if your horse dies, you'll just get the armor back, there's no survivability increase by carrying more, unless it falls somewhere inaccessible or burns up in lava.
Confirmed.