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My guess is that Gandalf and Pippin were referring to some Chess-like game played in Middle-earth. Remember that they were speaking Westron, not English, so 'pawn' and 'chessboard' are purportedly translations of the terms they used. Real-world chess is just one member of a large family of games - which includes not only related games such as Shogi ('Japanese chess') and Xiangqi ('Chinese chess') but also historical games such as Shatranj and Chaturanga (the ancestors of modern Chess). And further afield, there are many other families of abstract strategy games, both modern and historical - from the Germanic Tafl games to Nine Men's Morris to Go and its many variants. It would actually be surprising if Middle-earth didn't have some board-game from which an analogy could be made to warfare.
Incidentally, the bishop (along with the queen) was one of the last chess pieces to develop its modern name and rules. In Shatranj, it could only move exactly two squares diagonally, and could jump over other pieces like the knight. And it wasn't called a bishop - it was a 'fil' or 'pil', meaning 'elephant'. One can't help but imagine Hobbits playing a game with pieces called 'oliphaunts'.
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