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.
FLEECE, VELVET, MIST, and PILLOW all fit the category "Soft things". They suggest a gentle texture. 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
TOKEN / TILE / DIE / PAWN
TOKEN, TILE, DIE, and PAWN 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 you fold
LAUNDRY / LETTER / ORIGAMI / MAP
LAUNDRY, LETTER, ORIGAMI, and MAP all fit the category "Things you fold". A crease is part of the process. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things in a gym
KETTLEBELL / ROPE / BENCH / MAT
KETTLEBELL, ROPE, BENCH, and MAT all fit the category "Things in a gym". Training rooms are their habitat. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.