Let me start by saying that this sounds tricky.
It would be helpful to know what software created the backups. Without knowing that, you are working in the dark, and so some blind experimentation may be needed.
If these files are of any real importance, your first step is to make sure you have suitable backups. Otherwise, whatever program you use to restore files might change the files. Also, it wasn't clear how you're using the floppies: if the data is on floppies, or if you were just using them as part of a restore technique. Floppy disks can deteriorate with age and usage, so get at least one copy of the data onto modern hardware. And, if you really care about the data, make another copy of the data (which should be easier after at least one copy is on non-floppies).
Software like DOSBox might use standards, like 1.44MB disk formats in disk images. However, actual backup software from that era would sometimes do things in ways that were not quite so standard, like fit 1.76MB on a floppy. So I would suspect that using software like DOSBox might cause restoration software to not work as expected. DOSBox gets lots of reviews for being great, but it can't handle data right if it doesn't see a copy of the data. To maximize the chances that you have every critical "bit" of data, it might be necessary to create disk images of any floppies.
I've heard that the Backup and Restore programs built into DOS are specific to the version of DOS being used. So you might need the restore program from the same version of DOS, if that is how they were backed up. Even worse, I have heard that such programs were buggy.
What you are missing might be the first file in a series. Some programs would start numbering, starting with the second archive file that was created. For instance, ARJ files would use file extensions of ARJ, A01, A02, A03, etc. RAR files would use file extensions of RAR, R01, R02, R03, etc. (Actually, RAR had another pattern of some sort as well, depending on the version of RAR being used.) As for a program that just used pure numbers, I don't readily recall one, but I do think that too many programs used the pattern of just naming files after numbers, so that is not a telltale sign that can let someone just easily point to one specific program. Any specific advice may be a long shot that is more likely to be wrong than right. I realize this wasn't intentional, but the question ends up being fairly broad just because there are multiple possible answers.
Providing further details in an answer is probably infeasible without more details. Knowing the characters of the archive files may be much more helpful for humans to figure out the file type. There may be a pretty decent probability that even just the first 3 to 5 characters might be sufficient to give some key clues, while being rather low in likelihood of sharing damaging data. To figure out those bytes, you would want to view the files in a hex editor. That actually stands a good chance of providing some substantial clues on how to open the files. You might want to get permission to share the files-- if not to the whole Internet, then perhaps with a rather trustworthy expert who is bound by an NDA.
You mention that utilities are asking for a drive letter. It could be helpful to know just what those utilities are. Most of the more useful programs I've used to interact with archive files would typically take a filename. So, knowing what software you suspect may be helpful for people to agree, or rule out some ideas.
Since it seems you're desiring drive letters, make sure you know about mounting directories to drive letters. In DOS, SUBST can do that. In Windows, SUBST might also work there. Also, SMB file sharing can do that (by mounting a shared folder... this can be done using NET USE from the command line). In theory, APPEND or JOIN might also do similar things, but in actuality what I've read over the years is that those programs are probably more trouble/danger than help, so you probably want to stay away from those. (Things like operating system bugs, or differences between versions, may make them more of a liability than a positive use.)
I wouldn't put much faith in the TrIDNET analysis. I'm not familiar with the program, and so it might do a fabulous job. However, the answer that it is an old Norton file just seems like an answer that might be based on guesswork.
I know I threw a lot of advice your way. If you were looking for an actual clear-cut answer, it was the part about using the Restore program from the right version of DOS. But, I do admit that although the answer might be technically correct, it may admittedly have a somewhat low chance of success if you don't know what software that was.
You sound like you might just be looking for advice/direction/any-hints at this point. Well, in my opinion, the most promising approach is probably to look at the hex bytes and get a better idea of what types of files you're working with. Then you'll at least be in a far more informed position to ask some more specific questions.