I can't set osu! to my second monitor. The option for "fullscreen mode" doesn't give me a title bar when turned off, so I can't drag it to my second monitor. Alt + Enter doesn't window it either.
Ehm... do you have the "Letterboxing" option enabled by any chance? If that's the case, just disable it and you should be able to run osu! in windowed mode, drag it wherever you want, etc.