Even though the minecraft launcher shows that the custom skin is currently being used, the player will wear a default skin when playing the game. There are no error messages related to this in the launcher.
Edit: does not seem to happen when "use beta version of launcher" is checked off
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1.16.2

Huh. I just unchecked the box just to check, and my skin is still working fine. Probably was just a temporary glitch . . .
Maybe you should mark this invalid for now. Sorry!
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We're tracking this issue in MC-101579, so this ticket is being resolved and linked as a duplicate.
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That is very odd, there shouldn't be any difference between our beta version and the release version. Which version of Minecraft are you launching?