Perhaps I'm overlooking something.

I believe that what you described is correct.

even if that media contains an exact file/folder replica of your source media.

Haven't tested to see if it makes any difference but it could be that the case has to be the same also. The instrument OS is a linux variant that is case-sensitive. Do you have the same problem if that 'file/folder replica' uses the exact case?

Might be worth testing. Not sure why they would use an internal reference rather than an absolute or even a relative path.

I copied the files and folders exactly.... Didn't work. Only when I completely imaged the device did it work,I suspect because it duplicated the internal volume ID. Don't know if there's a way to reference the drive without using that ID in Linux.

I should clarify that my first attempt was to format a new USB drive on the target M, give it the same exact volume name as the source and then copy all the files/folders from the source to it. That didn't work. I had to make a sector by sector copy of the source device. 

I do know that Yamaha Genos keyboards don't have this problem and I think they might use a Linux-based OS. 

Interesting problem.  I don't use this feature at all myself and have the 1st gen Montage only.   Since .WAV files are only used by the "Audio Recording and Playback" internal function -- and structures aren't saved in any file -- it's difficult to "see" how these are referenced.   There's no real visibility except by the firmware developers as far as I can tell.

As far as I can tell, the live set audio files are referenced by a number from 1 through 128.  There must be a table somewhere else (internally) which matches up the audio file number (index) to its file properties (volume, directory, filename).   If there's no table then I wonder if the indexing is according to some traversal method of the directory structure.   It would be easiest to deal with if always saving files to the root directory since this would at least remove ambiguity in how files are ordered wrt the directories/sub-directories.

My guess would be when using a flat root-only USB stick (no directories) that the files would be in order of the directory entries.  1=1st entry, 2=2nd entry, etc.  And under most OS's this order would be determined by which was copied to the stick first.  And listing files in an OS from the command-line (without using ordering flags) would show the order of the original USB stick.   It's possible that even the volume name isn't important.

None of this is very user friendly - although this is typical Yamaha.  To present the underlying technical "rules" to the user without abstracting and place the burden on the user to deal with the consequences.  

Assuming most everything in this thread is accurate - it would be a great opportunity for Yamaha to enhance the elegance of Montage (M) by removing the requirement for users to worry about anything other than having the same wav files on a USB stick removing any dependencies on the order of files on the USB stick, volume name, size of the stick, etc.

I've submitted this issue to Yamaha support indicating that they've provided no guidance on how to restore a user file backup containing .WAV file links.