Wombats produce cube-shaped droppings because the final stretch of the intestine has walls that stretch unevenly — softer in some bands, stiffer in others — and that uneven squeezing molds the drying material into cubes before it leaves the body.
It sounds impossible until you look at the mechanics. Material moving through the wombat intestine dries out slowly as it travels, and in the last section the gut wall does not squeeze evenly all the way around. Some regions give and stretch; neighboring regions stay comparatively rigid. Over many rhythmic contractions, the softer material gets pressed against those stiffer boundaries, corners gradually form, and what started as an ordinary lump leaves as a recognizable cube.
Researchers at Georgia Tech, including Patricia Yang and David Hu, worked this out by combining physical examination of wombat intestines with mathematical modeling, publishing the findings in the journal Soft Matter in 2021. The shape is built in during digestion — the wombat is not doing anything deliberate with its muscles at the end.
Wombats have poor eyesight and rely heavily on smell to understand their world. Droppings are a message board: piles left in prominent places announce whose territory this is. A round dropping placed on top of a rock tends to roll off and get lost. A cube stays where it was put, stacks higher, and keeps the message readable for the next wombat passing through. The geometry is the point.
See the short version: "Wombats Poop Perfect Cubes — Things You Didn't Know" on LOE PRESSURE (Science #32).
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Sources: Yang et al. (2021), Soft Matter, on the intestinal formation of wombat feces; Georgia Institute of Technology press release on the study.