I have 3 profiles: Default (called Personal), Foo and Bar.
These are the icons for my profiles on Launchpad:

(I edited out the real names)
As you can see they have all different icons, but my configurations on every single profile are the same: round icon with an emoji.
The names are not the same also, the default profile was renamed to Personal but no Orion - Personal application exists and the Foo profile does not have the Orion - Foo suffix.
All profiles have the configured icon as configured in the settings, as well as the correcy name.
When I launch a profile it should open that profile correctly, sometimes it does not launch correctly and the default profile, configured as Personal, opens the default profiled used to open links, that is the Foo profile.
Version 0.99.127.3-beta (WebKit 619.1.1)
Sonoma (14)