Hey there!
Crossplay just keeps your saves from other platforms. It will not transfer any purchases, as those are handled by the platforms (in this case, PlayStation and Stadia), not the game.
Currently it is not possible to transfer purchases, and personally I don't think it'll happen any time soon, either.
-- Baldy
I am no Google official. My messages and opinions are my own, and should not be used to judge Google or their products. Silver and gold Guides are independent volunteers that like to help out on the forums.