Thanks, Stephen, but I still cannot resolve this though I have completed some of the steps you suggested. I have doubts about the Credential Manager thing you mentioned:
2. Reset Windows Defender Credential Manager: Corrupted credentials stored by Windows Defender can also cause this problem. Here's how to reset them: Search for "Credential Manager" in the Windows search bar. Open "Windows Credential Manager." Find any entries related to "Windows Defender" or "Microsoft Defender Antivirus." Click on each entry and select "Remove." Restart your computer and see if the login prompt persists.
When I go there, I see this:
Is something wrong here, that can be the reason I need to log in every time? As you can see, there are no entries related to "Windows Defender" or "Microsoft Defender Antivirus" at all. There's a line "No Windows credentials" - maybe this is the reason? And also, two Microsoft Account user accounts entries. The 1st of them shows my previous Email which I changed to the current one, and the 2nd one shows my current Email associated with my Microsoft account. Can this also be the reason of this mess? The thing is that I changed the Email on my Miscrosoft ID long ago, but the mentioned issue only started recently. And currently I have only 1 user account on my PC, I checked that.