The wallpaper function in-game doesn't do anything when playing via Flashpoint, but if you save your score and settings via the settings menu you can go to the game on this website: http://www.dokidokivisual.com/magic_of_stella/star_liner_magicofstella_2016.php?&small=0
on a browser that supports Flash and when you click the wallpaper button in-game on the website, you can access the wallpapers you've unlocked.