iCloud on the web logs out everytime (in Safari, Chrome, Mozilla)

Hello Apple community,

I use web browsers a lot during the day to log in to my iCloud account for my mail. However, whatever how often I save my passwords in the browsers, and how often I mark those browsers as trusted (I have two factor identification), I keep on being logged out of my account all the time. I do not use ad blockers in two of the browsers (Chrome and Mozilla) and in the other browser (Safari) I whitelisted icloud.com. Oh, and I accept third parties cookies from websites visited. It's driving me crazy! What can I do to ensure I stay logged in, just like I stay logged in to Facebook, Twitter, Linkedin, etc.?

Best, Sarah