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.
MUSIC, CHOCOLATE, SOAP, and SIGNAL all fit the category "Things with bars". Bars can describe each answer. 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
PAWN / DIE / TOKEN / TILE
PAWN, DIE, TOKEN, and TILE 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
Kitchen verbs
WHISK / DICE / FOLD / SEAR
WHISK, DICE, FOLD, and SEAR all fit the category "Kitchen verbs". These are actions from a recipe. 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 / HEIGHT / STONE
FREIGHT, TENSION, HEIGHT, and STONE 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.