What happens when you collect all the puzzle pieces in one puzzle?

Nothing by itself. You have to actively combine them to get the item or aninal in your storage.

You can, if you want, collect 8/6 if you want to.

Usually, like for the advent calender 2020, where you could receive 6 puzzle pieces for 3 different animal, you might consider to combine them earlier, just to have a chance to combine a second animal of the same species later, in order to start breeding.