I don't know the answer to your question. But I recently discovered the PWA addon for firefox, and it can accomplish most of what you are describing with the icons and launchers; however it's not profiles, it installs certain websites as apps in a new firefox runtime, and only allows one tab open at a time per pwa.
this post was submitted on 11 Jul 2023
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@ja2 I have had the same problem! Because of a separate issue, I had an entry for firefox and firefox with a specific environment variable, and it would randomly choose which icon to say my session was coming from.
I would say it's infuriating, but I think I use that word too much.