Not sure I understand. Why should the number of home runs per year, the number of goals per year, and the number of touchdowns per year have equal variance?
You probably wouldn't expect them to. But the same kind of Stein phenomenon holds under a much broader set of conditions, including arbitrary covariance matrices and arbitrary quadratic loss, (see e.g. [1]). It's a very general phenomenon!
I'm going to go out on a limb with my intuition, and hypothesize that the underlying premise of the method zeroes in on a common characteristic of games humans tend to enjoy.
A universe of people wouldn't find these games generally interesting, if they didn't present outcomes above a certain threshold of unexpectedness. The underlying rules of each game are tuned into the equipment used, and a balance is reached, where game play is fair, but still requires players to develop skills.
Because each sport adheres to the premise of capturing interest in players and spectators, they all present the same scoring tendencies, when aggregating and generalizing.
If you change the motive of the activity (mix games with non-games), and the artifact that represents success (mix freely tallied points with rare physical tokens or discovered evidence), so that the behaviors being compared are dissimilar, the predictions will become unreliable.
For example, when comparing "victories" across lawyers, geological prospectors, and sports players you probably would not be able to make predictions about all, by lumping each area's statistics together. A gold-mining prospector probably wouldn't encounter success in the same way a trial lawyer would, and neither would help you predict or generalize a hockey game.
But, an oil driller, a diamond prospector, and a gold prospector would likely compare, based on the geological goal sought. A forensic analyst, a private detective, and a trial lawyer might compare, also, based on the human factors of investigation. And, thus, so too, with sports where freely tallied points measure a player's skill at achieving an event in game play.
[1] https://projecteuclid.org/euclid.aos/1176345691