If we are talking old purely about the C# language and ignore the surrounding ecosystem of available libraries, tools, and OS "compatibility", then it doesn't matter whether you are on Windows or Linux.
I don't have enough experience with using .NET Core for creating desktop GUIs on Linux.
On Windows, WinForms was just sort of kludged into .NET Core by Microsoft, they were originally going to just let it wither on the vine of .NET Framework. Microsoft is putting more emphasis on it's other GUI frameworks.
For game development, it really depends on the game engine (and/or game framework) you use and how well it works on the platform of your choice.