Hi all, so, the problem is 90 percent solved. It was indeed the problem with the graphic mode, just as Stefan's crytal ball told. Orinially the kernel driver should automatically detect, which mode is possible and then switch to it. But it doesn't. So the screen went blank.
The fix was easy: I added the parm "vga=791" to the kernel and the screen went to the required solution. Everything is working now, when I boot the stick on another computer. Thus I can say: generally it is working! However: When I boot this stick on the same computer, where I built the iso, it hangs at boot. I suppose, this is related to, that I am using encrypted filesystems on the harddrive. It appears (although the ISO is working completely in RAM), it wants the encryption password. This is strange. In the past I gor a solution for it: Before building and creating the ISO, I moved /etc/crypttab away and then created the ISO. This was working well in the past. You should know, that I have written a little script, which prepares the system (by deleting history and other unnecessary things or moving some things away), then build and at last move everything back. This is working like a charm. Oh, and maybe the issues I described might appear with a newer kernel, because KALI linux was running 6.1 but now it is running 6.5. However, this might not matter, just want to mention it, if someone else is interested and fell into the same issue as me. As I said: the problem is 90 percent solved, now I am just try to solve the (very!) little thing with the password issue. Stay tuned, I'll be back. Best Hans