How do the Skylanders figurines work

The data is stored in the figurine itself. While you play on the PC/Console versions, the data is updated continuously (that's why the figure must stay on the portal while in play), and the Skylanders keep their level when transferred between portals. The only exception is when playing on the 3DS version, where the data is stored in game and only transferred when the Skylander is transferred again into the game.

Also, the Skylanders top out at level 10.

I figured it out myself:

You have to sign in with a user, otherwise the game will not save (but will not warn you that it is not saving).