You can right-click the pinned taskbar icon for the profile and then right-click "Opera GX Browser" in the menu that pops up, then goto "properties" and look at the target field on the shortcut tab for the command that's used. In short, Opera uses the same launcher as the main profile but just with a command-line switch to tell the launcher what profile to load.
See https://www.reddit.com/r/operabrowser/wiki/opera/gx_profile_default_browser/ for an example way of setting a specific GX profile as the default browser. You'll see where the ico file for each profile is at also.
An alternative is to ignore Opera GX's profile feature and just use a standalone installation for each extra profile you want. Then, you can use a utility like method 2 at http://www.winhelponline.com/blog/register-firefox-portable-with-default-programs-in-vista/ to set a certain standalone installation's launcher.exe as the default browser. You'll just have to edit ApplicationName in "HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\FirefoxPortable\Capabilities" to say "Opera GX Something" (or whatever you want) so it doesn't say "Mozilla Firefox, Portable Edition" in Default Apps/Programs.