There is no battery on the mainboard, and I can't recall if my scores were saved (I only owned Puzzle Bobble 2).
If there is any sort of score saving I'd imagine it's on a per-cart basis, since you can swap carts and it would take lots of space to save multiple game scores on the motherboard, or simply lose them when a game is swapped.
Open one of the carts up and have a look for a battery?