iCloud for Windows repeatedly asking me to log in

Ever since the last iTunes update, iCloud for Windows is repeatedly popping up to ask me to log in again.

I enter my password (and I go through the 2-party identification and enter the access code), and it logs in. Then, after some time - sometimes, hours, sometimes minutes, it does it again. This repeats over and over.

When I log into my Apple account online, under "Devices", I see my PC listed multiple times.

This sounds like some kind of bug in the latest version that does not seem to recognize that I am logging in using the same PC. How can I fix this? My wife's computer has the exact same problem. We are on Windows 10.