Hi Lennart Schlotter,
Welcome to Microsoft Community. I'm Mosken, let me help you.
According to your description, I knew that your PS4 controller was recognized as a wireless device when you connect it to Windows 10 with cable, and it wasn't recognized by the game and DS4 Windows properly, even if you have tried updating drivers and plugging and unplugging the device, disabling/enabling the device, but still failed. Is that right?
The first thing to be clear is that your device was identified as wireless device depends on the properties of your device itself, not just how it's connected.
Then have you tried connecting the controller to another computer or connect it to the current computer via Bluetooth? This can help determine if the problem is the system or the controller itself.
Then you mentioned that you have tried disabling/enabling the device in the device manager, now I suggest you try in the device manager to try uninstalling the device first, disconnect, reboot the system. Then open the DS4 Windows and connect the controller by wire. Under normal circumstances, the system will automatically find and install the appropriate driver.
Or you can try to reset your controller by following the official steps provided by your controller manufacturer, or ask them for further technical support.
And you mentioned that you have tried to update the driver. If you updated it in the device manager, you can try to download the driver of the corresponding model of your device from its official website, and then install it manually.
The above steps are provided based on the information I have learned so far, and hope that could help you a little. If the problem still exists, please provide us a full PSR from the beginning, through the whole process, and until the error message is displayed, so we can further investigate.
Best Regards,
Mosken-MSFT | Microsoft Community Support Specialist