Tested also in a new browser with no cookies/cache. I can login sucessfully to any site and meta site of the web, but whenever I try to login to any chat, it autologins with a different user https://dba.stackexchange.com/users/11558/jynus instead of https://dba.stackexchange.com/users/30545/jynus (this user). I logged-in through Google authentication.
I tried to delete the old one, but it was no solution, logging out and logging in from the chat doesn't work, as as soon as I login, it authenticates with the new user. If I log out directly to the chat, with some of the links provided in other answers/login support text, it works (if it is the first login ever) temporarily, but as soon as I log into a room, I get changed to the other user. As a result, I cannot talk even in my own created rooms.
I've read the support page thoroughly, no success (posting here was the last resort).
Tested on Firefox and Chrome.