Why does Saturn have a hexagon and no other planet does?

The best answer is a standing wave in a jet stream. Spin fluid in a tank with a faster-rotating band inside it and the boundary between them can settle into a polygon - laboratory experiments have produced triangles, squares, hexagons and heptagons depending on the speed. On that account Saturn's six sides are just which polygon fits its particular winds.

Why only Saturn is the harder half of the question. Jupiter spins faster and has jet streams too, and has nothing like it. The likely difference is that Jupiter's polar regions are a chaotic cluster of cyclones - Juno photographed eight arranged around one at the north pole - while Saturn's polar jet is a single clean band. What decides which arrangement a gas giant ends up with is not understood.

The structure is about 30,000 kilometres across, has been there for at least the forty-odd years since Voyager photographed it in 1981, and extends far down into the atmosphere. Inside sits a storm with a genuine eye some 2,000 kilometres wide and eyewall clouds moving at around 500 kilometres an hour - a hurricane over a pole with no ocean anywhere to feed on.