I think I remember Pokemon blue saying that mew gave birth on feb 6th
Yes today is mewtwo's birthday as stated from the Pokemon mansion journal. Source: http://bulbapedia.bulbagarden.net/wiki/Mewtwo
Yes, it actually is. I was actually seeing this on Smogon.
A journal quote from the Pokémon Mansion states that Mewtwo's birthday is February 6. - Bulbapedia
