FWIW I agree with this guy, I can log in fine now, but skins are still borkin'. Me and my friend show up as Alexes.
Yep, me and friends on my realm have this bug so much that we call it THE chat lag.
/tell works, and so does /say. Interestingly, anvils don't work while chat is lagging, and signs get set to blank (permanently) every time the chat lags. I have a theory that this might be caused by the game's chat filter or something that checks user input, but I have no clue.
Using a VPN does not fix the skins issue, nor does clearing my ./assets/skins folder.
Edit: The issue is gone for me now.