I replicated this bug in the latest snapshot (24w21b) and it removed the custom name of the saddle. As a side note, this bug does not affect horse armors in my testing.
Have you tried this bug in Vanilla 1.20.6? I didn't managed to replicate it.
I wasn't able to replicate the bug
I replicated this bug in the latest snapshot (24w21b) and it removed the custom name of the saddle. As a side note, this bug does not affect horse armors in my testing.