The Reason The Star Shape Doesn't Look Like A Real Star
When starlight hits the atmosphere, it's knocked off course like any other wave — radio waves, sound waves, and so on. Atmospheric heat, pressure, particulates, and the like bend and distort the light to differing degrees in a process referred to as "atmospheric scintillation," as Sky at Night explains. This makes the star's pointy arms — a visual impression of its light — looked stretched out, or blurred. Light from farther objects is distorted even more, and appears more blurred. This is also why closer objects don't blur so much.
As for the number of star arms, rays, points, what have you, it depends on the human eye. As MinutePhysics goes into on YouTube, the number of rays depends on how many "suture lines" a person has in the lens of their eye. Suture lines are lines where pieces of the lens fused during development. Four suture lines means four rays of a star. Five lines means five rays, and so on. We can see the same phenomenon in cameras. A camera with four lenses shows points of light with four rays, a camera with six lenses shows points of light with six rays, and so on. This creates the "diffraction pattern" that an individual sees when a star's light strikes the eye after passing through the atmosphere.
In a really cool twist, each person sees a different pattern. Each human eye develops unique suture lines, so each person sees stars differently.