Since you already know that your program allows multiple instances, of it, open at the same time, then there ought to be a way of getting those multiple instances open using just the shortcut icons in your Startup folder.
However, just having slightly renamed icons to the exact same program file is probably not going to work, as you have already found out. The reason is that the "Target" line (which is really a command line) of the shortcut icon is still identical in each shortcut icon there.
For example, if you open the Properties of one of those shortcut icons, and select the "Shortcut" tab, you may see something like this:
"C:\Program Files\SomeCompany\In House System\OurService.exe"
You may need to ask someone, in your company, as to what parameters are needed to follow that command line to open up a single directed instance, and then include that, slightly altered, for each Shortcut icon in your Startup folder.
For example, suppose the command line needs to have a file named after it, then the Target line might look like this:
"C:\Program Files\SomeCompany\In House System\OurService.exe" doc1.svc
and another instance, for the second shortcut icon then might be:
"C:\Program Files\SomeCompany\In House System\OurService.exe" doc2.svc
and another might be:
"C:\Program Files\SomeCompany\In House System\OurService.exe" doc3.svc
etc.
Or, (again after verifying with the developers of your in-house service software), it might require a "switch" (which usually is preceded with a slash "/") after the command, such as:
"C:\Program Files\SomeCompany\In House System\OurService.exe" /newwindow
and subsequent Shortcut Target lines might use that exact same Target (command) line.
OR! perhaps it might be a combination of both, like:
"C:\Program Files\SomeCompany\In House System\OurService.exe" /newwindow doc1
then
"C:\Program Files\SomeCompany\In House System\OurService.exe" /newwindow doc2
Check with the developers of your software. I'll bet it can be done.