It actually happens all the time in other leagues. Atlanta Hawks used to be St. Louis Hawks and they won NBA title there. So did Kansas City Royals, who then moved to Sacramento. I think Braves won World Series in Milwaukee and Boston.
The Kansas City team you're talking about had been the Royals in Rochester and Cincinnati, but they changed their name to Kings when they moved to Kansas City because of the existence of the MLB Royals.
In MLB, in addition to the Braves (1 in Boston, 1 in Milwaukee), you have the Dodgers (1 in Brooklyn), Giants (5 in New York), Athletics (2 in Philadelphia, 0 in Kansas City), and the Senators/Twins (1 in Washington)
In the NBA, you have the Royals/Kings (1 in Rochester, 0 in Cinccinati, 0 in KC), Hawks (1 in St. Louis), Lakers (5 in Minneapolis), Nationals/76ers (1 in Syracuse), Warriors (1 in Philadelpha), Supersonics/Thunder (1 in Seattle)