Problem identified

Once a Media has been uploaded, the Media Manager does not allow to move it to another folder.
Just like it is possible to Rename an Image via the interface, it should be possible to Move an Image via the interface.

Note: actually, there is a little hack: if you rename for example joomla_black (.png) which is in the root folder to banners/joomla_black then it will be moved to the banners folder. But you can only move this way to a subfolder, not to a parent or sibling folder.

Proposed solution

On top of the existing possibilities like Rename, Edit etc, add a Move to option