On macOS, instead of running Firefox via the app icon, I run a an AppleScript[0] that forces Firefox to launch with the Profile Manager window. I have it bundled as a one-step Automator application, which I run from Spotlight but which can easily be added to the Dock or whatever.

I used to have this script built as an app bundle via Platypus, but that stopped working somewhere between Yosemite and Mojave (likely due to increasingly strict OS file permissions or something). The Automator app doesn't seem to suffer from this issue...I think I just had to grant it a couple permissions the first time I ran it.

I wasn't able to figure out how to set a custom icon from within Automator, but it's not too hard to do if you're comfortable mucking around inside app bundles. The icon file is stored under <app-bundle-root>/Contents/Resources/, and its name (sans extension) is referenced in <app-bundle-root>/Info.plist.

[0]: https://gist.github.com/ilikepi/9d2e17e0d3b3efd6fc0584f46f09...