The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
MOUSE, COIN, COMET, and KITE all fit the category "Things with tails". Each can have or refer to a tail. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Board game pieces
TILE / PAWN / TOKEN / DIE
TILE, PAWN, TOKEN, and DIE all fit the category "Board game pieces". You might move or roll these on a tabletop. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Night sky
VENUS / METEOR / NEBULA / ORION
VENUS, METEOR, NEBULA, and ORION all fit the category "Night sky". Look up after sunset. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Book parts
CHAPTER / SPINE / INDEX / COVER
CHAPTER, SPINE, INDEX, and COVER all fit the category "Book parts". They help make up a printed book. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.