It's not incorrect to say "shiniest", but it sounds much more natural to use "brightest."

Shine is more commonly used as a verb than as an adjective. "That one is the brightest star shining in the sky."

The correct way to say it is, "the brightest star". Perhaps because we think of stars as things that emit light, whereas shiny describes things that reflect light. This might be the reason why "the shiniest star" sounds odd.

No, they aren't. "Brightest" is correct and natural. "Shiniest" is wrong, or very strange. In talking about the light they give off, we say that the stars, the moon, the sun, etc. "shine." But, oddly enough, we do not say they are "shiny," so we cannot talk about "the shiniest." The word "shiny," for some reason, always refers to reflections from polished or glossy objects, like polished metal. The noun, "a shine," also refers to polished objects. The sun "shines," the light of the sun can be called "sunshine." But we do not say the sun is "shiny," and we do not say the sun "has a shine," because nobody has polished it. "They polished their shoes to make them shiny. All of their shoes had a brilliant shine. Joe's shoes were shiniest of all." "In the hot weather, their faces were shiny with perspiration. Poor George's face was the shiniest of all."