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I played the Steam demo, then bought it on GOG. Due to the demo being Windows only and I playing on Linux, I had to copy over the save files from the proton path. The issue is that while I can continue the save from the demo, all unlocks from the achievements are reset to 0 and I have no idea where the data for that it saved to ferry over.

In which file is that data saved?
This question / problem has been solved by Ville Mönkkönenimage
The achievements are kept in wherever Unity keeps it's player preferences file. In windows that's somewhere in the registry file, and I have no idea how Proton would keep the registry. Here's the documentation https://docs.unity3d.com/2020.1/Documentation/ScriptReference/PlayerPrefs.html
You could probably copy the achievements that way, but that would be a bit of an effort.

If you're looking to get the achievements to your GOG profile, you won't get them this way, because the game only sends them to the server once each achievement is first received.