The Therapist Gamer

I played this game on the PS Vita, but's it's also on the PS3. There really is no way to get a good feel of this game simply with screenshots, so sorry. You play as a series of shapes, but I guarantee that if you play through this game you'll be more attached to these shapes than any you've encountered before. They each have strengths, weaknesses, and a personality to which you'll probably relate.

Basically, it plays as a story-driven puzzle game - and it does well on both counts.