Betting on matches of chess is much different. In poker you have a betting round after every move, the betting size matters and the betting is one of the main sources of information you get. In chess even if you bet after every move the amount wouldn't matter and you get little to no extra information from the bet.
Take all the pieces off the board and now the bets align with the information asymmetry. The only thing exclusive to poker is the hierarchy of the combinations of cards.
