Saddles lose their NBT data when equipped on horses, zombie horses, skeleton horses, mules or donkeys via right-clicking

The bug

Saddles can be equipped on tamed horses by right-clicking on them while holding a saddle. However, doing this erases the saddle's NBT. This issue does not occur if the saddle is equipped via the horse's inventory.

This also affects zombie horses, skeleton horses, mules and donkeys.

How to reproduce

1. Summon a tamed horse:

   /summon horse ~ ~ ~ {Tame:1}

2. Give yourself a renamed saddle:

   /give @s saddle[custom_name='"Test"']

3. Right-click on the horse while holding the saddle
   → The saddle is equipped on the horse
4. Inspect the saddle equipped on the horse
   → The saddle does not have its custom name