I have witnessed this bug countless times. These are the exact instructions that worked for me, after many tests, every time:

1. Make sure you have the latest version of GOG Galaxy first. The installer checks whether you have the latest version. At least it's good on that.
2. Close the app completely (Ctrl+Shift+Esc, filter processes by "gog", close all processes related if there are any)
3. Remove cached and login data, the fastest way is by typing Windows+R, typing "%LOCALAPPDATA%/Gog.com" and then removing the Galaxy folder you'll find there. Also, empty the Recycle Bin afterwards.
4. At this point, you should have a fresh clean app... Try to log in now! However, I found instances where it /STILL fails (the f****ng motherf***er!). Instead of typing your password on the login screen, click on "Change user". This does the trick because it forces the app to remove any remaining personal data... now it doesn't know who you are: it will work now. It did for me every time at this point.

I'm writing this just so I can help anyone else. I run GOG Galaxy sparely and my intuition (I'm a software engineer) tells me the app doesn't handle some edge cases well. I think there's probably a missing flow/algorithm to handle valid but outdated auth tokens.

I hope this helped you!

Question:
If it's so much trouble, why bother with it?

Reinstall Edge : it works for me !