I have a dual setup, both digital monitors. Using XP Pro. Old Matrox video card
If I want to move Stellarium to my second monitor I press F11 while running on screen 1 to reduce the size on screen, and then just drag it across to screen 2.
I can have Sellarium running on both screens at the same time, one daytime the other night time
Seems to work OK, even on my relatively old PC.
Good luck.