I ended up solving it by fiddling with some settings (Show Advanced button):
Settings/Web/General/Allow Fallback to Insecure Connections: changed from 'On same network as server' to 'Always'
Settings/Server/Network/Secure connections: changed from 'Preferred' to 'Disabled'
Settings/Server/Network/List of IP addresses and networks that are allowed without auth: just removed my local LAN and left this field blank
Settings/Server/Network/Enable HTTP Pipelining: unchecked it
After changing this and manually restarting Plex Media Server processes, by running 'systemctl restart plexmediaserver', the error went away.
BUT: after changing every setting back to their defaults, and restarting plexmediaserver, just to reproduce the problem and locate wich specific setting was the culprit, the error had not returned. 8-O
My guess is that something was changed in Plex Web Client Version 3.20.5 that messed with authentication. Even after logging off and on again with my Plex username the problem kept happening. But only with Chrome on Windows. Unfortunately I did not have the chance to find out specifically which setting "solved" it.
Things are still not 100%, as they were previously with version 1.7.*, every now and then I'm getting errors related to 'can't play this media' or something on Chromecast. Despite the errors, I have been successfully streaming movies to my Chromecast v1 from Windows.
I hope someone else could investigate this further.