Start with the plainest picture the two words give you: a game, played with a ball. That covers football in a park, tennis on a court, and children throwing something back and forth after school. Two things come with that picture and never leave it. There are rules everyone has agreed on, and there is a result. American speakers narrow the picture to one match that a crowd has come to watch. Then English takes the same picture off the field: a situation you are in can be described as the game you happen to be playing, with its own rules and its own final score.