since I make and listen to music, my external sound card "doesn't" work on Ubunut Mate 22.04.01.
I'm not surprised. A lot of manufacturers "forget" to create drivers (or deliver specifications) for linux.
I've carefully chosen my external soundcard for Linux support:
8 channels XLR input (10 jack output) and via the backplane coupled to another 8 XLR in ( 8 XLR out) linked via ADAT-lightpipe (all XLR inputs support 48V phantom power):
Works with Ardour (via JACK audio connection kit) and with Audacity (via the internal portaudio drivers)
So the only way is connected to the display via headphones and the sound from the graphics card (HDMI).I have DP and HDMI for the displays, at first I thought it could be the HDMI or DP, they both probably support at least 16 bit and at least 192 kHz sample rate. Unfortunately I somehow have no subs, no bass on the sound with monitor headphones (Beyerdynamic DT-770)
The problem is not in the digital domain: Headphone amplifiers in displays are much too weak to drive the Beyerdynamic DT-770. This is also true for headphone amplifiers on most motherboards. This is the reason why you have distortion when turning the volume up and why a lot of bass is missing.
(Another possibility for missing bass is a faulty ground contact in the mini-jack plug connection, resulting in a 'left minus right' signal in both headphone speakers but let's not go there yet)
Any ideas ?
Yes: if you want to continue using the weak amplifier in the display, connect the input of a separate headphone amplifier (or an old hifi amp) to the headphone output of the display. By having a much higher input impedance you will most likely get your bass back and you will have enough power to drive your DT-770 
However: Even if you have a separate headphone amp installed between display and headphones, don't expect "high-end" 16bit/192kHz quality from your display audio output. The analog part in display sound outputs are pretty mediocre to begin with.