MSI has the concept of multiple instance installations using product code changing transforms. These transforms can (but don't have to be) embedded in the MSI. Here's some background reading.
http://blog.iswix.com/search?q=multiple+instance
Read the IS12 then the IS2009. Expect many broken links as Microsoft destroyed their sites. Here are 2020 links.
https://learn.microsoft.com/en-us/windows/win32/msi/installing-multiple-instances-of-products-and-patches
https://learn.microsoft.com/en-us/windows/win32/msi/authoring-multiple-instances-with-instance-transforms
https://learn.microsoft.com/en-us/windows/win32/msi/installing-multiple-instances-with-instance-transforms
When using embedded transforms you are limited to whatever ProductNames you choose when you built them. You could also dynamically generate transforms and apply them.
WiX's standard bootstrapper application doesn't really have much in the way of supporting all the scenarios I talked about in my blog articles. Other tools such as InstallShield and AdvancedInstaller have more.
As much as I like WiX, and MSI for that matter, this is really advanced stuff that takes a long time to understand, develop and test solutions for. You might want to look at using a tool such as InstallShield or Advanced Installer or simply creating a normal straight up MSI that installs a template of an application and then create your own custom instance management tool that allows the user to spin up multiple instances of your application.
I'm an WiX/MSI/InstallShield/DevOps consultant. To be honest, this would not be an inexpensive project if I was to quote it out for a customer.