I'm having the same issue. I go to https://www.icloud.com and click on "Sign In", enter my user name and password and enter my 2FA code.
For about 2 seconds I see a "Welcome ape_ape" and I'm then redirected back to https://www.icloud.com and I've been logged out. Clearing cookies doesn't help, using a different browser doesn't help. Using a different computer doesn't help.
I've checked the system status site and everything has a green dot. The system status site is useless. Name a social network and you'll find people complaining about this issue. It's literally blowing up, but all systems are green. What's the point of a system status page that can't tell you your stuff is having problems?
Thankfully logging into community support doesn't suffer from the same issue, otherwise I'd be unable to post this and about ready to throw all my Apple devices into the garbage.