Making a semi-realistic Kirby Alternate Universe (AU) here. Popstar's shape is a result of a member of this powerful race using secret magics to alter reality in this AU.
What would its day/night cycle look like? What would the shadows from the points look like? Would there be locations that never or rarely receive light?
In light of some hard thinking and looking at the comments, I've decided that the canon popstar is too anomalous. I've decided instead that it is a normal round planet with five enormous mountains, angled like so: