Why can't I move files from my Ubuntu desktop to other folders?

The reason you cannot drag files is due to limitations in the support of desktop icons. Since a few Ubuntu versions, icons on the desktop are not anymore provided by nautilus file manager (aka "Files"). Instead, a Gnome Shell extension "Desktop Icons" has been created for that. This extension is rather new and still quite limited in its capabilities, in particular with respect to the interaction with the file manager.

This has happened because Gnome Shell does not support desktop icons. They consider the desktop not to be the appropriate place for folders and launchers. Most of the time, it is covered by your applications. Instead, users are encouraged to place frequently used launchers on the Dash, and launch applications from the Application overview. For files, users are encouraged to work from Files, or from within the applications, rather than placing a range of files in a single Desktop folder.

Consequently, at some point, Gnome developers removed the code to provide desktop icons from the file manager. Therefore, Ubuntu used an older version of nautilus file manager in Ubuntu 18.04, in order to continue providing desktop icons. Nowadays, they ship with an up-to-date version of nautilus, and support desktop icons through the Gnome Shell extension. Hopefully, with time, the extension will improve in replacing the former functionality of nautilus with respect to icons on the desktop.

@vanadium has provided an exhaustive explanation of why we can no longer drag and drop to and from the Ubuntu desktop. (+1) I would only like to add that if you open the Desktop folder in Nautilus you can drag and drop anything to and from that and any other folder that is open in a separate Nautilus window. It’s the best workaround that I have found so far.