iCloud keeps logging me out when accessing Mail via browser

Hello, highly respected community!

A problem with iCloud.

I have a working AppleID which I'm able to sign in on my various Apple devices, from an old iPhone and Mini to a iMac with Monterey. ID works fine and lets me manage itself. It sends 2FA codes and even allows me to 'trust a browser'. But when I try to enter iCloud services such as Mail in a browser (Safari or Brave) it just keeps throwing me out, back to login screen.

Pretty straightforward actions lead to signout. I have reset my password and even logged off AppleID on all devices except iMac but it didn't helped. What's happening with it?

[Re-Titled By Moderator]