I would also love this feature.
In addition, the behavior described in the manual, Entry Editor - JabRef, does not seem to be the one in Linux either.
In Linux[1] the Shift modifier moves to the entry directory and renames if there is no other file of the same extension in the entry directory. (Maybe related to Rename multiple files for one entry ? )
The Ctrl modifier, and using no modifier, copies to the upper level directory and does not rename if other files of the same extension exist in the destination directory.
Using Shift-drag, therefore, will not rename the file if a pdf already exists in the entry directory. But files of any other type will be renamed if no other files of that same extension exist.
In addition, adding multiple files to an entry in one go (i.e., selecting a bunch together) does not seem to be possible.
Edit: I just noticed there is a github issue for moving/copying without renaming: Move/copy a PDF file for an entry WITHOUT renaming (drag and drop) · Issue #7313 · JabRef/jabref · GitHub
Thank you,
Ramon
[1] At least when using Dolphin, or Konqueror or pcmanfm under Xmonad.