And you have a profile whereas in Mate you don’t have one. So Mate is wrong despite the fact that you have proper colors. Meanings that you fallback in sRGB which is ok for your screen (by chance). So now, would be nice to know which profile is used as “display profile” on Darktable under Cinnamon.
Another option is that the Cinnamon profile is wrong/broken !
The colour distortion is caused by that auto-generated profile (you can see it’s auto-generated, because it has edid in the name).
Somehow remove the EDID profile from Cinnamon or install sRGB as your display profile (I don’t use Cinnamon, so can’t help you there), OR, as a workaround, set darktable to treat your display as sRGB. I’ve already posted the link, it’s the ‘soft proof’ one from Cinnamon desktop color shift - #28 by kofa
That won’t fix the setup of your computer, ‘only’ cripple darktable so it ignores the bad profile. You can also cripple Firefox by disabling colour management. ICC color correction in Firefox - Mozilla | MDN
In the future, please ask and investigate before claiming something must be a bug in darktable.
Another option (the best option if you ask me) is to create a proper profile for your display using DisplayCAL for example (you need a hardware like Datacolor Spyder).
Firefox seems to use the system profiles by default (even for other monitors!); this matches my experience that I talked about the other day. Same thing for me: a profile was auto-generated and I did not know about it, and it provided really strange results (too warm, in my case).
As pointed out by Kofa, there are about:config options in Firefox to stop it from loading that profile, but the most radical approach is of course to disable the profile system-wide.