"Blue Mario" is a recolored Mario character, first appearing in Mario Tennis for the Nintendo 64. He is not playable, but he can be seen sitting in the referee's chair during some matches when one is playing as Mario. "Blue Mario" can also be seen in the multiplayer game of the Super Mario Advance series, but relation to the "Blue Mario" of Mario Tennis 64 is not explicit and thus currently unknown unless confirmed by Nintendo. It is also currently unknown if he is related to Mario or not.
As an unnamed background character, and a recolor at that, "Blue Mario" has had no official role or importance to any Mario game in canon, and there is no known information about him. It is uncertain what his name is, what his life is like when he isn’t acting as referee for Mario, what his relation to Mario or friends is, or what he’s even like as a person and a character. Because of this, there have been many theories about him, his identity, and relation to the Mario cast.
It has been claimed that his color scheme is meant to be a reference to the original Mario design in terms of color, but citation is needed - and it may just be a coincidence. However, it is true that Mario acting as referee - and the alternate blue colorscheme that serves as the origin of "Blue Mario" - is a reference to Tennis for the NES in which Mario is the referee and happens to wear blue. But this does not change the fact that this means that "Blue Mario" physically exists separately from Mario.
There have also been cases of those who have theorized that "Blue Mario" was an early version of Waluigi. However, this is clearly false as Mario Tennis 64 was the same game that introduced the existence of Waluigi as a character and as an evil counterpart to Luigi. "Blue Mario" can still be seen in the referee's chair during any match starring both Mario and Waluigi.