When entering offline mode multiplayer server both 1.18.2 versions just randomly show custom skins or not depending on luck.
If custom skin didn't show, player can (without closing game) go into single player, skin will start working, then join server again and it will show there too.
Modpack in question is "Create: Arcane Engineering" if that's important.
Currently I'm trying to turn off different mods to see if something influences this, with no luck so far.

@zlainsama
I incidentally figured out a way to overcome this issue! When I run the game with special JVM params to block Microsoft auth servers, the issue just disappears. Maybe has something to do with official account sync? Not sure, but here's the workaround:
https://github.com/ivxvm/hyperlauncher/blob/d3c226a5b051525a0e816a71396ae4e5ba1eaa3a/src/modpack.py#L94