THE WINNING TEAM shows one of the most common of anachronisms: the leading lady wears her hair in the fashion of when the movie was made, instead of according to the fashions of the period of the story. Through the entire film Aimee wears her hair in the short fashion of the 1950s, even though Alexander died in 1950.
Not only does much of the stock footage (as well as regular footage) show players wearing numbers on their uniforms long before this was the case, a number of the crowd shots were actually shot in the late 1940s or early '50s, as indicated by the clothing and hairstyles.
The fans are shown giving a loud, enthusiastic standing ovation to Alexander for his pitching heroics in the 1926 World Series. However, the games that Alexander won in the 1926 World Series were played in Yankee Stadium, meaning that the fans were rooting for the Yankees, not the Cardinals. The fans were stunned when Babe Ruth was caught stealing at second to end the Series - not just because of the way it ended, but because their team had lost to a team it had been heavily favored to beat.
The film refers to Alexander beginning his career with Galesburg in the "Three Eye" League (Illinois-Iowa-Indiana League), a "B" league at the time. Galesburg and Alexander were actually in the short-lived Illinois-Missouri League, a "D" league. The following year (1910) he played for Syracuse in the class "B" New York State league before the Phillies bought his contract.
In Game 7 of the film, the Yankee batter, probably Bob Meusal strikes out for the last out. In the real game, Babe Ruth committed the last out by being caught trying to steal second base. It is the only time a World Series ended by a runner being caught stealing.
In Grover Cleveland Alexander's rookie season, which was 1911, Alexander's wife (Doris Day) is talking with the wives of other players. She asks who the opposing pitcher is, and she's told, "That's Eddie Plank. He won twenty games last year." That year would be 1910, and Plank only won 16 games.