I know that the brother's have been said to be twins, which means they were born on the same day. According to the Mario wiki, the source for their October 11th birthday is an Issue of Nintendo Power Magazine. I personally believe that Mario doesn't have a confirmed Birthday unless it's confirmed by being in a game. I'd be fine with their birthday being October 11th, but I just don't trust that a single magazine issue would be considered important enough as a source to be completely safe from being changed at any point if Nintendo prefers.
If you'd like an out of universe answer to the question, Mario first appeared in Donkey Kong on July 9, 1981 and Luigi first appeared in Mario Bros for Game & Watch on March 14, 1983. That's neat because it would make Luigi's birthday Pi day.
If you'd like an out of universe answer to the question, Mario first appeared in Donkey Kong on July 9, 1981 and Luigi first appeared in Mario Bros for Game & Watch on March 14, 1983. That's neat because it would make Luigi's birthday Pi day.