When you set a custom skin with a classic or normal model it is "saved". However, when you open the game again regardless of whether you are connected to the internet, or be logged in to a Microsoft account. The character model changes, for example:

If you applied a custom skin with a "classic model", then you close and open the game again. The skin model changes and is set to "slim model"

For proof of the bug, check out this video: https://drive.google.com/file/d/1MQYG7DOxILC5Ltzk2fW1AdZEhuWlnuvw/view?usp=drivesdk