Short update: I managed to boot the GPi Case 2 and Recalbox without any problems. To be honest, I don't know what the reason was that it didn't work. I didn't really do anything different this time, except that I used the dock on the first boot. I don't know if that was really the problem. The fact is, everything works as it should, even without the dock.
I put the micro SD card back into my PC after trying to boot my GPi Case 2.
Maybe the hardware.log will help?
[ 4.51] case/manage.py(mainInstall:218): Script info - case: , previousCase: , installedCase: , phase: 0, previousPhase: -1
[ 4.53] case/manage.py(DetectGPiCase2:40): trying GPi2
[ 4.54] case/manage.py(DetectGPiCase2:80): gpi2: compute module 4 found!
[ 4.64] case/manage.py(is_hdmi_connected:71): gpi2: error reading card1-HDMI-A-1 status: [Errno 2] No such file or directory: '/sys/class/drm/card1-HDMI-A-1/status'
[ 4.64] case/manage.py(DetectPiBoyDMGCase:131): Trying PiBoy DMG
[ 4.79] case/manage.py(DetectPiBoyXRSCase:138): Trying PiBoy XRS
[ 4.95] case/manage.py(mainInstall:247): No case detected
[ 4.95] case/manage.py(mainInstall:248): Processing case none on phase 0
[ 7.25] case/manage.py(mainInstall:218): Script info - case: , previousCase: , installedCase: , phase: 0, previousPhase: -1
[ 7.25] case/manage.py(DetectGPiCase2:40): trying GPi2
[ 7.26] case/manage.py(DetectGPiCase2:80): gpi2: compute module 4 found!
[ 7.36] case/manage.py(DetectPiBoyDMGCase:131): Trying PiBoy DMG
[ 7.50] case/manage.py(DetectPiBoyXRSCase:138): Trying PiBoy XRS
[ 7.65] case/manage.py(mainInstall:247): No case detected
[ 7.66] case/manage.py(mainInstall:248): Processing case none on phase 0
[ 78.92] case/manage.py(mainInstall:218): Script info - case: , previousCase: none, installedCase: , phase: 1, previousPhase: 0
[ 78.93] case/manage.py(mainInstall:248): Processing case none on phase 1
Okay, that's weird. Yes, I used the Raspberry Pi Imager for flashing. I select the correct model (Raspberry Pi 4 - Models B, 400 and Compute Modules 4), the OS (Emulator and game OS - Recalbox, Recalbox - Raspberry Pi 4/400/CM4/GPi Case 2/PiBoyDMG), my micro SD card and flash it.
The GPi Case 2 is for CM4 only.
Thank you very much! Hopefully they'll find out what's wrong. 
No, when I docked the GPi Case 2, the d-pad doesn't work either. The whole situation is really weird, because the installation of Recalbox is more than simple, and it worked flawlessly months ago.
Thank you for your answer. Yes, I'm pretty sure I'm using the right version, the handheld one.
What I forgot to mention: If I try to boot the device without the dock after installation, nothing happens. But when I have docked it and connected the HDMI cable to the dock, I can see the EmulationStation menu, but the controller does not work, so I can't navigate through the menu.
Hello!
Recalbox would be my first choice, but it does not boot after a problem-free installation on my micro SD card. A few months ago I bought a micro SD card, installed Recalbox as indicated, inserted the micro SD card into the device, started it and everything worked without any problems. Now I have bought a new micro SD card, installed Recalbox in exactly the same way, but now I have the problem that the device no longer boots. The screen of my GPi Case 2 remains black. It is not due to the new micro SD card, as the error also occurs with the old one, which worked without any problems a few months ago.
I then installed RetroPie as a test to see whether it was due to Recalbox or perhaps to my Raspberry Pi. With RetroPie, however, the boot worked without any problems. So I was able to rule out that it was due to my Raspberry Pi.
Does anyone have any ideas on how to solve the problem? I really want to use Recalbox instead of RetroPie, but Recalbox doesn't seem to want to be used by me. 