The players's currently applied skin renders as a thick-armed model on singleplayer worlds and multiplayer servers, regardless of how it was configured in the launcher's skin customization section.
Restarting the game or the launcher multiple times does not help, nor does applying a different skin (which happens to be taking longer than normal to be applied at the time of writing), switching between thin-armed and thick-armed skins, or resetting the skins back to the default Steve model.
User is signed in and connected to the internet with full access, the launcher reports being connected, and has no issue confirming the applied skin and correctly rendering all skins in the skins section of the launcher.
(I have included some screenshots of what I am referring to. I make my own skins, and always leave the appropriate areas magenta-coloured on the skins intended to be thin-armed models, so that if they are rendered as thick-armed, it is glaringly obvious, as demonstrated. In order, the first 3 images are of how the skin renders in-game, and the final image is how the launcher relays it to be.)