I've now reinstalled my computer - I wanted to start fresh with the installation so that everything really works smoothly and no remnants from the Jaguar (I skipped Panther) cause any trouble. So I made backups and reformatted and set up the box. Everything went well. Dragged music via drag-and-drop into the iTunes folder, that worked too. Played the first purchased piece - I have to authenticate my computer. Hello? What? I'm using it on exactly the same computer I bought it on, but I have to authenticate myself?
DRM is simply an insult to adult customers.