@waylow I’ve just found the answer myself here: You need to run the script again every time you start Blender. But in my case, it’s not necessary because Blender seems to execute it automatically. If I remember correctly then there’s a setting where you can decide whether Python scripts shall run automatically or not. I think one important reason for that is that you don’t want to let Blender run any harmful script hidden in a downloaded .blend-file . Another possibility according to the posting mentioned above is to install that script as an addon in order to have it present with every Blender start.
Also worth mentioning that this script is only visible in the “Item” panel of the “Pose Mode” which is logical since it’s meant for making rigging easier but people can easily be confused if they dont notice that they aren’t in “Pose Mode” and look for the “Melvin Rig UI” for example in the “Item” panel of the “Object Mode” where it doesn’t show up.