[MC-258360] Horse armor loses its NBT data when equipped on horses via right-clicking

The Bug:

Horse armor loses its NBT data when equipped on horses via right-clicking.

It's important to note that this issue does not occur when equipping the horse armor through the horses' inventory. This problem is most noticeable with dyed leather horse armor; when right-clicking a horse with dyed horse armor, the horse armor loses its color.

Steps to Reproduce:

Summon a tamed horse by using the command provided below.
```
/summon minecraft:horse ~ ~ ~ {NoAI:1b,Tame:1b}
```

Give yourself some leather horse armor that contains custom NBT data by using the command provided below.

/give @s minecraft:leather_horse_armor{display:{Name:'{"text":"MC-258360","italic":"false"}',color:11141290}}

Right-click on the horse while holding the horse armor to equip the horse with the horse armor.
Take note as to whether or not horse armor loses its NBT data when equipped on horses via right-clicking.

Observed Behavior:

NBT data of the horse armor is lost.

Expected Behavior:

NBT data of horse armor would not be lost.

Video:

https://www.youtube.com/watch?v=2htKzz-VOXc