RESOLVED: Sometimes the solution is... just in front the nose. For years I used as my login name my iCloud email address (like mynickname@icloud.com) and as it was for long long years, I thought it's my primary login name - and it always worked quite well. But in reality my primary login name is my private email (like myothernickname@gmail_or_yahoo.com) that I used when I created my iCloud account (sometime in mid of 2010's).
So the solution (at least for me) is: DO NOT LOGIN with your iCloud email alias (something@icloud.com) but LOGIN with your primary account login name (your_original_login_name@otherdomain.com). And now iCloud app for Windows works!
And it was all the time in front my nose: when I was logging to iCloud via web browser, I could notice that primary login name (not alias) was written below my name...
What I understood, Apple has changed recently somehow it's terms and conditions and the one and only way to login is via primary login name, not via any iCloud alias.
Hopefully this will help you.