Okay, I finally got it!
The problem was that the steam overlay needs to get injected before D3D/OpenGL are initialized. For a reason it is possible to test the steam overlay function directly in a debugging session through visual studio, but this does not count for a windows project of MonoGame.
The solution for the windows version of a MonoGame is, to compile the project and then launch the output (.exe) through the steam client. For debugging purpose you can add the compiled exe to your steam client as a "Non Steam Game" and then launching it by clicking the play button in the steam client.
The steam overlay should be successfully injected now. This can be checked by opening your task manager and searching for the process "gameoverlayui.exe" (it's from steam).
I created a new branch called "Windows" with the newest commits in it. At the moment I have a branch for DesktopGL projects (master) and Windows projects (Windows). I will create a better solution later.
Further I will work together with @throbax on this repo to add more interesting and meaningful features to it like "SteamAPI.RestartAppIfNecessary()" to ensure that the game will be launched through steam (soft DRM and also to ensure that the steam overlay will be correctly injected) and "SteamAPIWarningMessageHooks" for receiving warning messages from steam (debugging).