trusteft: None of the games sold on GOG have keys. Not sure what you have there.
Zandolf: I bought it here. I purchased the game and most of the DLCs from Slitherine. When it came to gog, I bought it again because it was the "Gold" version and on gog, no keys. I wanted to avoid the game keys for each DLC when I reinstall them. But, now the thing ask me for a key. and I think there was a work around back when gog first got it. I purchased it on gog. Sept 13, 2021.
I installed it for the first time few days ago.
It never asked for any key.
Perhaps your Windows installation still had info from the previous installation and it messed everything up.
Try to uninstall them all, delete them and perhaps even do a registry cleaning of the game if they are still there after uninstalling it. Then try to reinstall the GOG version.