Lucian If I turn the phone screen of, then the monitor also gets black.
Yes, because you locked the phone and it's in mirror mode. Whatever your phone shows is shown on your monitor.
Lucian The resolution is only 1080p
This is a software limitation in AOSP that may be resolved in the future. Pixels with stock firmware (and I think generally all Androids) have this too.
Lucian Not all of the monitor is used, for some reason it has black areas.
Yes, because you're using the same aspect ratio as your phone screen and your monitor adjusts to preserve it. Taking up the whole screen is only available in desktop mode which is currently unstable and entirely different from what you're using right now.