There are more than three songs per game, at least for some. That said, if the devs are to be believed, the game card is filled to the brim with content, so unless they cut out a few FF games, then there's no way there could be more songs per game (other than choosing a bigger game card, but considering how unprofitable handheld games can be nowadays and the quite non-enthusiastic response to Theatrhythm's announcement, opting for a more expensive bigger game card (+ the extra dev time costs) probably didn't seem too lucrative a business decision).
While it would've been nice to have more songs, what number of songs would be enough? We've got a series for which many entries have soundtracks that have closer to 100 songs, enough so that you could make the usual size music game or even two out of one FF game's music. At what point would they have enough songs for people to not go ""I wish it would have more songs from each game, XX just isn't enough". Also, even though the number of songs per game isn't too high, the number of songs in Theatrhythm is still more than in most other music/rhythm games.