If you're looking for what to do about it, skip directly to the bold section.
So, I've had a "nice" week of trying to sort this out with Adobe Customer Services to no avail.
First I spent six hours with two customer service reps remotely stopping services, uninstalling CC App and cleaning up, removing folders, setting access privileges and reinstalling the CC App in various ways and a number of times over, with and without antivirus active.
After that we had two working pieces of software with residual copies of every previously installed Adobe software but Creative Cloud application not recognizing them.
Numerous messages to Adobe Customer support -- which apparently is a robot that sends the same link based on message content rather than a person -- with the response each time essentially being telling me to do the same exact thing that didn't help.
So I had to uninstall everything, which was the last thing I wanted to do with a whole slew of After Effects plugins that would take several days to reinstall IF I could find an installer for After Effects CC 2015 to begin with.
So, today the sign-in problem returned with no access to any Adobe programs, not even trials.
A number of variations of uninstalling and reinstalling later, including the suggested handling of creating a new user account on the computer NO improvement.
Until I googled the question in Finnish and found that I should try the following:
1) Log out of CC App
2) Rename two folders "SLCache" (C:\Program Files (x86)\Common Files\Adobe) and "SLStore" (C:\ProgramData\Adobe)
3) Log back in
That took two minutes after which everything worked like it should. Apart from the fact that I no longer have any of the applications installed.
Would have really been nice if Adobe Customer Service had told me this at any point.