The packaging is misleading. Technically, yes, there are 20 games on the v4, but your character can not play all 20 at the same time.
There's the basic 6 that you get from the Game menu, Jumping Rope, Mimic, Shape, Dance and Flag. Then there are 14 different jobs that your Tama can get when it grows up. The thing is, your Tama can only have 1 job at a time, so you only get 1 extra game. If your Tama doesn't get married and have a baby in a few days, you'll be offered the chance to get a new job, so for awhile you can play a different 7th game.
Hope that clears it up!