The closest game I have come across for teaching sentence structure was a set of words that were magnetized on the back and divided into nouns, verbs, and inflections. Unfortunately, I only saw them in Estonia, and that a dozen years ago.
But taking that idea, a teacher could make word-cards and deal them out or distribute them in some way and then have (elementary) students form sentences by playing like “Go Fish.”—“Does anyone have a verb in the past tense?” or “I need all your adverbs” etc. The other advantage to board or action games when teaching on a higher level—say, 6-8th grade—would be cards with prefixes, suffixes, inflections, etc. and use them to show how base words are altered by them—I remember a Latin teacher in high-school in the 50’s who used to take common English words in groups and strip them of their prefixes, etc. to show the Latin root (and therefore helped us see cognates)—such as “incorporate,” “corporation,” “corpulent,” etc. to show the Latin word “corpus.” Today there must be hundreds of video games such as “Word” that show language structures. The term “games” implies contest, sides, prizes, visual combinations, a sense of accomplishment, etc., so a teacher should be careful to keep the learning part in the activity and to make sure the “game” has structures and goals (the opposite of tetherball, for example). One teacher in undergraduate college made a game out of building a highly visual Chomsky tree of the Pledge of Allegiance, the Declaration of Independence, etc., as a class group activity for display at a school Open House. One final note: Remember diagramming sentences? That’s a structure “game.”