Thanks for the info everybody! I will look into setting up a VM as I prefer the desktop app to the browser. But that also might be overkill and using multiple browsers might be the more reasonable approach.
sorry I'm late to post here, but just in case anyone else is coming here looking for the same answer.
I've tried replicating the Discord.app in the Finder and renaming it to the account I want to use, and I've also tried the aforementioned Terminal command. Neither work as I expected because of the fact that Discord.app is one of those stupid Electron applications that is actually a Chrome web browser with a fancy skin. When you have one instance of Discord.app running, you are actually not using that binary directly. Discord.app is for the lack of a better word, just a launcher. If you open up Activity Monitor and do a search for Discord you will see various instances of Discord Helper running, in particular Discord Helper (Renderer). This is where most everything is getting done.
So when you launch a copy of Discord.app, it goes to launch the Discord Helper apps and sees that it is already running and effectively quits itself.
So, yeah, unless you hack the code to open different instances of the Helper apps, this isn't going to work; it's not going to work anyway because every time Discord auto updates itself, these changes are going to get destroyed.
So, so, so yeah, as someone else mentions you're going to have to use some sort of virtualization to run a separate OS for each of the accounts that you want to run in Discord at the same time. I think a huge waste of resources on your computer.
The best way to really do this and still have access to things like your shared clipboard to copy between the multiple accounts is to use a web browser that has the ability to segment the cookies like the new Safari's Profiles, Firefox's official Containers plug-in, or a Multiple account extension in Chrome or Chromium (which probably is best because it's a Chrome app). Then you can keep them all open at the same time along with your Discord.app.
It would be best if the official Discord community comes up with an ability to do this, but I don't see it happening.