I started having the same issue a couple of months ago. I am being logged out of Windy website both on desktop (Windows+Firefox) and mobile (Android+Firefox) every few days. 1h/3h setting is also forgotten.
No, the cookies aren't set to clear on browser close;
No, I am not running in private browsing mode;
No, it does not happen with other websites, they stay logged in without problems.
Is it possible there was some update to the site backend recently that introduced this bug? Is it possible that the login session became somehow dependent on the cached files rather than cookies (because these do get cleared periodically)? Has there been any progress on this bug, which has been reported by multiple users already?