@lubos: For the record, I probably found out why the pop-up appeared and what I could do about it.
I did a search and it seems that the wording of the macOS pop-up is a bit misleading. It shows whenever an app asks access to a location to which it has not obtained explicit permission. This includes data from other apps, but it is not restricted to that.
Recent versions of macOS are rather strict. You have to give explicit permission to apps to access most areas outside of their own control. This includes Desktop, Documents, external drives and others.
So after I gave macOS permission through its settings to allow Manager access to external drives, the problem seems to have gone away.
I guess it is a side effect of the - very handy - option to easily change Manager’s data folder. Before there was no need for macOS to ask for permission, since the data was normally all within Manager’s own folders.