Eric, You are awesome!
I have a similar problem. I finally got my PC running WIN 10 to sync Outlook 2016 and iCloud. It took 5 app specific passwords and as many installed on the iCloud for Windows app. A step that was just not happening, was the connection of the iCloud mail, contacts, calendar, etc eve though the iCloud for Windows would run 'successfully'. I run an update of Office from inside my Outlook 2016. After a restart and re-install of he iCloud for Windows, unchecking the mail completely and waiting a few seconds, the connection process worked, the TAB appeared in Outlook2016 and all works fine after some fine toning.
I shared this as I expected something similar would solve the same problem of MAIL but nothing else for on iPhone and iPad using the Outlook for iOS app. No such luck. Deleted the app on the iPhone, re-installed and set up all my email accounts. I am not touching my iPad with this again until the iPhone is working properly. I do not have syncing for contacts, calendars on my iDevices with Outlook for iOS.
Any Ideas? It is not as simple a needing to get an app specific password for each device to use to log into the iCloud account in the Outlook for iOS app. Apple support by phone did not seem to understand the problem and after escalating me that tech needed to do research but never has called me back (appointment was set for Saturday). I am not so happy. But I am in search of a solution to this. This Outlook for iOS app is promising and worked well until mid June when everything went upside down for our non Apple developed apps.
Postings here always miss the details of the situation I am trying to fix or show confluence between the Windows desktop Office/Outlook and the iOS and add MAC Outlook. At times you cannot tell what folks are talking about. So, I have separated my discussion of what finally worked for my desktop and my iDevice woes.