In the history of MLB. When considering the major yearly awards, those being: Rookie of the Year, Most Valuable Player (MVP), Cy Young, Reliever of the Year, and Manager of the Year. There is exactly one instance of both of the winners having the same name in the same year.
In 1980, the Orioles Steve Stone won the American League Cy Young Award. Over in the National League, future Hall of Famer Steve Carlton won the his leagues Cy Young award. Marking a year of Steve supremacy on the mound.
There were a couple other close calls however:
While their names are said the same, the 1966 Rookies of the Year Tommie Agee and Tommy Helms have different spellings to their names, disqualifying them from this illustrious accomplishment.
Additionally, all the way back in 1925, the Washington Senators Roger Peckinpaugh won the American League MVP, and the Cardinals Rogers Hornsby won the NL MVP. However, that pesky "s" changing both the spellings and pronunciations of the two names keeps them out.
The only other remotely close calls involved Manager of the Year award wins by Tommy Lasorda and Tony La Russa. Which may not be all that close, but I think is neat.