Thanks to everyone who assisted me with this issue. I figured out the problem and now have a working installer.
The problem was that when you add a folder to Inno Setup, it adds the files but doesn't put them in that folder; it puts them with the executable. This means that the content files were in the same directory as the executable, not in a content folder. To change this, I simply had to add the directory to the Content folder destination path in the Inno Setup code file.
I think I might try to make a tutorial on making a Monogame installer with Inno Setup to help anyone else who runs into the same issues as I did.
Thank you to everyone who helped, and hopefully this will help anyone else out there with any issues!