Leather horse armor has no provision for an overlay texture like human leather armor has. This overlay present in the human armor allows for a texture with no tint index to be layered over the texture that takes the dye tint index. This has been present as such since leather horse armor was first added in the 19w08a snapshot.

This is clearly apparent in how it dyes the entire texture when the blanket and harness pixels on the texture are given hue and saturation. If these pixels were meant to be dyed, they would not be given any hue. I can only assume that it was intended to be coded to have the overlay and this single texture version was a placeholder that ended up never being replaced.

Horse armors, left to right: diamond, gold, iron, leather, leather dyed cyan.

Closeup of leather and leather dyed cyan, showing how entire texture is given tint index with no overlay.

Leather horse armor dyed cyan with saddle, showing strong differences in saddle and armor harness textures.

Comparison to human armors, leather and leather dyed cyan, showing clear overlay.

Leather horse armor in resource files, showing no overlay texture while human leather armor contains such textures. Files were extracted from 20w48a (because I was too lazy to do another extraction), while screenshots are from 20w49a.

Leather horse armor texture, clearly showing greyscale areas meant to be given tint index next to pixels intended for an overlay texture.

This is a duplicate of MC-144493, which was incorrectly marked as Resolved as it could not be reproduced (while evidence of it being fixed was never shown), however the OP has not come back to verify the status in over a year so I am making a new ticket. OP did not provide any images.

What I expected to happen was...:
See the leather horse armor have a dyed portion and undyed portion.

What actually happened was...:
The leather horse armor was dyed across the entire model.

Steps to Reproduce:
1. Dye a leather horse armor.

2. Place on a horse.

3. Observe.