A while back Sean added the new "Is birthday?" feature on Game Finder searches to find games where it's a given player's birthday. For example, here are the 55 guys since 1920 to get 4 hits in a game on their birthday. Actually, it's 52 guys since Yastrzemski, Vina, and Colavito did it twice.
Anyway, since the Game Finder searches also allow us to group for most such players in the same game, we can find games with the most different players having a birthday on the same day.
In 2010, there were 4 games where two different players had the same birthday:
It's kind of annoying...the results don't make it easy to see which players had that birthday. But if we go to the birthday page for the given date--for example the September 16th page--we can figure it out pretty easily. In that game it was Brandon Moss and Chris Carter.
This is an intensive search and we can't do many years at once, so here are a few games of interest I found from recent years.
On May 20th of 2009, the Reds and Phillies had a 3-birthday game: Jayson Werth, Ramon Hernandez, and Adam Rosales.
In 2003, there was a 3-birthday game: Rays/Jays on August 5th (Carl Crawford, Eric Hinske, and Bobby Kielty).
In 2001, the Braves and Marlins had 3 birthdays on September 24th (Jesse Garcia, Kevin Millar, and Bernard Gilkey).
And here are some more 3-birthday games:
Giants/Astros on 9/13/2000
Indians/Jays on 10/3/1999
Braves/Pirates on 5/10/1944
Giants/Braves on 9/6/1938
I couldn't find any 4-birthday games going back to 1920.