I've been having the same problems. I used a method to get a custom cape/elytra for my character and I thought it was that, but this only happens when it's loaded in the 2nd or 3rd slot. All my other skins are still working
I could be wrong, but I'm pretty sure you have to use a USB stick to get it on the other Xbox. Minecraft worlds aren't saved to the cloud, they're saved to the Xbox storage so it won't appear on the other Xbox even if you are using the same account.
I've been having the same problems. I used a method to get a custom cape/elytra for my character and I thought it was that, but this only happens when it's loaded in the 2nd or 3rd slot. All my other skins are still working