Shiny Plushies are little Pokémon statues with the shiny palette that you can place in your claim. Simply right-click the plushie where you want to place it. To pick it up again, just right-click it while crouching.
This will give you 6 Shiny Crane Game Tokens to use at Shiny Crane Game at /warp GameCorner
Plushies in the Crane Game are randomized. You can see all possible options by left-clicking the button on the machine.
Too many plushies in a single area can be laggy, please limit the amount of plushies within visible distance to under 150. If you have more than that, you can split them up in multiple locations, as long as there's a few chunks of distance between them.