Thank you very much for your prompt and informative reply. I appreciate it.
The 396 books that were recognized by CC before were all indeed epubs, the others are mostly pdfs. Haven't paid attention to that before.
I have solved the problem by following means: I moved the books from the folder "Books" on my device to a new folder I created especially for that purpose. Then I started moving the books back in again, but not all of them. Instead I moved them in packs of a hundred or so. Then CC scanned those smaller packs. I did get interrupted a couple of times due to "timeouts", but the job's done.
Now I can actually start using Calibre Companion
Thank you all for Calibre, Calibre Companion, and all the support and updates. Life's just nicer with software like this.