The region your game uses is dependant on what game version you're using.
For example; if you want to have a green line under your name and play with European players, you should use the PAL version of the game.
Blue line - Americas
Pink line - Korea
Red line - Japan
White line - Taiwan/CTGP
You can change your country very easily with AnyRegionChanger, but it only affects the flag and position of your Mii on the globe in the Wi-fi menu.
I am aware there are codes that can do this but I haven't seen them, nor have I used them.