Confirmed, this is the exact issue I am having on Xbox One and Android but with official Mojang skin packs from the Marketplace.
I can indeed confirm this as a bug on the Android version. The skin stays the same until the app is closed, then the selected skin reverts to the default Steve or Alex, thus I have to change the skin choice yet again to play as it in-game each time now and it doesn't save the skin. This is even occuring with the Minecraft Skin Pack 1 which is an official pack from Mojang.
Edit at 3:33 PM CST: I have also confirmed this as a problem on Xbox One.
Yes, I can again verify that I've been on 1.13.0. It's the Minecraft Skin Pack 1, and the skin says Limited Usage which is incorrect.
Edit: I should clarify that this has been an issue specifically on Android and Xbox One since the 1.13 update. Version 1.14 for Android seems to fix the problem with skins resetting.