Try running darktable-cmstest and see if it can pick up your display’s custom profile. You can put a copy of the file in ~/.config/darktable/color/out (you may need to create that directory) if you want to explicitly select your custom display profile inside darktable, otherwise by default darktable will use the system profile it picks up from colord.
$ darktable-cmstest
darktable-cmstest version 3.2.1
this executable was built with colord support enabled
darktable itself was built with colord support enabled
primary CRTC is at CRTC 0
CRTC for screen 0 CRTC 2 has no mode or no output, skipping
CRTC for screen 0 CRTC 3 has no mode or no output, skipping
CRTC for screen 0 CRTC 4 has no mode or no output, skipping
CRTC for screen 0 CRTC 5 has no mode or no output, skipping
DisplayPort-1 the X atom and colord returned the same profile
X atom: _ICC_PROFILE (967156 bytes)
description: IPS231 #1 2020-08-25 17-12 2.2 F-S XYZLUT+MTX
colord: "/home/froggy/.local/share/icc/IPS231 #1 2020-08-25 17-12 2.2 F-S XYZLUT+MTX.icc"
description: IPS231 #1 2020-08-25 17-12 2.2 F-S XYZLUT+MTX
DisplayPort-2 the X atom and colord returned different profiles
X atom: _ICC_PROFILE_1 (0 bytes)
description: (none)
colord: "/var/lib/colord/icc/ASUS PB238 #2 2019-12-23 18-12 2.2 F-S XYZLUT+MTX.icc"
description: ASUS PB238 #2 2019-12-23 18-12 2.2 F-S XYZLUT+MTX
DT picks them both up. Is there a benefit of setting up DT vs letting KDE manage it?
Better to keep DT set to “system display profile” rather than forcing darktable to a specific icc file, so that if the window is moved to your other screen, darktable will pick up the right profile for that screen.
One thing I notice is that the X atom is not correctly set for your second display. In that case, it may be safer to go into your darktable settings, and on the “miscellaneous” tab, under the “interface” section, set the option “method to use for getting the display profile” to “colord” instead of “all” (I’m not quite sure what the system does when the X atom and colord settings aren’t in agreement, so maybe just tell it to use the colord info, which is showing the right profile).