Chess is a great visual spatial game. Of course, it has elements of sequencing, which is often difficult for the natural visual spatial learner. But it is spatial by nature. Each piece on the chessboard has meaning because of its context and relationship to every other piece. It is holistic by its nature. Any one change in a chess board, probably changes the entire meaning of the configuration. Expert players have the ability to see many steps or sequences in advance. But the sequence is meaningless if it is not mated with the ability to "see" the whole board and every relationship on it.
Almost every sport is a fantastic visual spatial game. Some involve more linear thought and sequencing than others. But you're hard-pressed to find a sport that's not spatially intensive. A common spatial element in sports involves the perception of space and time. Frequently the space and time involve the calculation of speed as well. You can see an example in a baseball outfielder. The outfielder sees a ball high in the air and has to predict its trajectory and often run to intersect the trajectory, not just get there before the ball lands. It's a very complex set of calculations. Yet good athletes can do it almost instinctively. They practice doing such tasks, because the practice teaches them, not just physically, but mentally, how to do those kinds of visual spatial tasks.
Ping-pong can be an amazing feat of visual spatial acumen. There are no computers and no machines that can acquire information, process it and respond as fast as a world-class ping-pong player. It is remarkable in large part because of the tremendous accuracy required in light of the very high speed and frantic pace of high-level play. Like most types of visual spatial learning, first players learn the underlying concepts such as how ball spin will affect a variety of shots. After they gain mastery of a concept, they practice faster and faster. With greater practice, they can respond with visual spatial understanding through impulse; they no longer have to think about how they should respond. They can, instead, intuit the right response much faster.
These examples are mostly multiplayer games designed for recreation. There are other games that aren't designed to be done as recreation. They are specifically intended to be done for exercise. You might call them "brain teasers." Many examples can be found in cards or other exercises where the player looks at multiple shapes and discerns a pattern, then arranges the pattern in a spatially logical order. Many of these games or sets of games are not just exercises but tests to try to understand a person's style of thinking and aptitude for different types of thinking.