Thank you BDAquq, just the last confirmation I needed! I saw that one of the Macintosh HD - Data disks had the House icon and I also tested that that was the one, where newly created user files go. It is the ‘hidden’ volume in Macintosh HD - Systems.
I found also that I had to do something - not only because of the impact on Time Machine: as the system is confused, some of the newly installed applications landed in the additional, wrong partition! And then of course, don’t show up in Finder Applications, even though functional. And the iCloud Drive backup (when I turned it off) also ended up in the wrong partition.
So I moved the data that were created there since upgrade to my Desktop. Apart from the Applications, as that will not work. Then I unmounted (and therefore erased) the additional volume without the House icon for the user folder with the Disk Utility. Installed all lost applications again and all is working fine as far as I can tell, including Time Machine and all Applications, showing up in Finder.