hi for better understanding take a look to “Advanced Session Plugin” by mordentral (Advanced Sessions Plugin - Programming & Scripting - Epic Developer Community Forums)
you get the gamesessionname by the onlinesubsystem when you create a session or as a sessionresult when you search for sessions. for steam it is equal the steamname…take a look at multiplayer shootout sample
so if you want to use custom servernames you have to add a custom session property (Online Subsystem Types in Unreal Engine | Unreal Engine 5.2 Documentation) and use that info in your server browser instead of using the gamesessionname