This isn't really related to the birth thing, but I just had a thought. If we postulate that the princess's full name is Peach Toadstool (her letter in SM64 seems to indicate that), and also accept the quasi-canonical belief that Mario and Luigi's last name is Mario (based most explicitly on the movie, but also drawn from the name "Mario Bros."), then the reason Toad says "Thank You Mario" in SMB even when you're playing as Luigi is that everyone was on a last-name basis at that point, and as they got to know each other better, they started using first names (of course, you really couldn't tell with Mario).