Based off of the DS' implementation of friend codes, you can play with random people (and in the case of Clubhouse Games it removes the drawing/chat and instead just lets you pick standard questions/comments as well). The only thing friend codes are for is for playing with your friends, not making it so you cannot play with anyone else.

Of course, you can select to play with only those who you are friends with, but that does not prohibit you from being randomly matched up with an opponent who you have not exchanged codes with.

The only game that I know of that you had to have a friend code for was Animal Crossing, and that merely because the average person wouldn't want random visitors trashing their towns.

Games that allow random play:
Mario Kart
Tetris
Clubhouse Games
Metroid Prime Hunters

I'm sure I'm forgetting a few more, but those are the ones I have.