IMHO (please anyone correct me) :-&
The $'s are used by the backup/restore program to indicate that the file is part of a backup set (but not compressed) or the files are part of an install disk set. I have definitely seen this kind of thing before. You may simply have to rename the files by finding out what the $ originally was. Renaming will work unless the files are also compressed. See if you can find a "disk 1" to locate any DOS batch files that were used to restore the files ("insert disk 2", etc).
I have also seen $ in front of partially recovered files, but why would you have disk after disk of the same partially recovered files? That does not seem to be a logical assumption. I am guessing that you have a stack of install disks.
The actual extensions could be (I am making an educated guess)
$CO, $DB, $NI, $CI, $AI, and $PI
ICO, MDB, INI, PCI (?), AAI (?), and DPI (?)
Anyway, The restoring program looks up the extension using some sort of retrieval table and copies to the restore destination with the $'s replaced by the appropriate character from the lookup table.
The actual program that created the backup is unknown to me, but I bet that it's a "major" vendor install or archive tool software of the time like Norton or Symantec or Fastback or something. Looks like a late 80's - mid 90's. Back then software companies used to rename their files so that a non hacker would have to use the install program to install software onto a harddrive, and force a register step somehow.
Educated speculation.
For kicks, start up a Windows 3.1 or 3.0 system and try this from the DOS prompt:
cd \ [enter]
dir *.?CO /s [enter]
...this should output all of the files that have an extension with the last two characters of "CO." If you get a match, then you have a candidate for what the $CO is when renamed during install. Repeat with each of the $nn extensions. If you really want to be thorough repeat on different IBM PC or clone from the era of 1985-95
If you have to de-compress a file try LZH.exe if you can find a copy. THis is a pretty flexible de-compression utility of the era. newer de-compression programs would be another option.
Bill D