This time something different :)
For some reason since weeks I had problem to connect to O365 mailbox. Initially, it stopped work so I decided to remove mailbox from Outlook (2016) and to add it again but that did not work. For whatever reasons, no matter, what I tried Outlook was not able to connect to O365.
Yesterday, it started to be even worse as when I went to SharePoint Online to test processing of documents all was ok and when I tried to open documents in Excel Online that worked too but opening of files through normal version of Excel was not working.
Surprisingly, Excel was telling me that I am already logged from some other machine which was not the case and additionally any Office desktop application was showing error next to username.
By opening this I saw: "Error - Sorry we can't get to your account right now. To fix this please sign in again." and logically, I tried to sign in again but that did not work so I was thinking "great, now beside Outlook, rest of Office apps are not working".
Of course, I did sign-out and then again sign-in but no progress. I even checked my user account under Windows Settings => Access work or school but that did not help.
Ultimately, solution was to go to Office app (I tried Excel) and to sign-out and then to go to Windows Settings => Access work or school where I clicked on Disconnect below my O365 account. After doing restart (not sure if this is really needed) I went to Excel and this time I logged again without any warnings and then I tried to add again my O365 account to Outlook 2016 and that worked.