Jet_Black: This has now happened to me. Did either of you find a solution?
To add a bit more info: My integrations seem to be working but the sync just hangs. I tried scanning my GOG Games folder and it told me it detected no games, and to top it all off, every time I run it it picks 3 random games to display them as being in my library but no others.
I didn't find a solution but was invited to open a ticket in the GOG Galaxy Jira. I did that and supplied some files from my GOG Galaxy directory.
I had to reformat my PC in the meantime (because of other reasons) and then it worked again. But before doing that I think it is worth trying to remove GOG Galaxy and reinstall it again. Perhaps this will help.