(I didn’t put this in Game Room because it’s more of a general musing than a serious discussion topic). Lots of field/court games involve a ball. Tennis, racquetball, football, water polo, volleyball, basketball. (A few notable exceptions such as frisbee golf, ice hockey.)
So why is only baseball referred to as a “ball game”? You never see people on the 18th green saying, “That was a hell of a ball game.” Announcers at Wimbledon don’t say, “This is the most exciting ball game we’ve seen in months.” But baseball is always going out to the old ball game.
It helps that the name includes the word “ball,” which tennis and golf do not. “Ballgame” is an understandable abbreviation of “baseball game,” but it would not be so for a tennis match.
Secondly, baseball is simply older than the other sports that do use the word “ball.” Baseball was a well established professional sports enterprise when basketball, volleyball and raquetball were INVENTED, and football as it presently exists (in the USA) developed after baseball, too. So baseball got the name first.
Not universal by any stretch, but Jeff Fisher always referred to a Titans football game as a “ballgame” in any context where “game” might have been uttered by somebody else.
It doesn’t sound that odd to my ears. Here’s a recent example of it being used in speech (as parts of quotes in the story.) There are three uses of “ballgame” in that story, from two different people.
Just search for “great ballgame” and “basketball” and you’ll find tons of examples. I used the phrase “great ballgame” because “ballgame” on its own was giving me generic “basketball is a ball game” type of results.