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.
PAWN, DIE, TILE, and TOKEN 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
Things with crowns
KING / TOOTH / TREE / WATCH
KING, TOOTH, TREE, and WATCH all fit the category "Things with crowns". A crown can sit on each of these. 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 / INDEX / SPINE / COVER
CHAPTER, INDEX, SPINE, 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.
Category
Hidden numbers
FREIGHT / TENSION / STONE / HEIGHT
FREIGHT, TENSION, STONE, and HEIGHT all fit the category "Hidden numbers". Each word hides a number sound or spelling. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.