i don't think they meant to take it as far as making Birdo ''trans'' considering the year she debuted.
It was more of a joke character at that time.
Nowdays Birdo is treated as a female and just like Yoshi is a species and is full of males , Birdo is also a species now full of females.
Let's not forget the SMB2 manual doesn't treat that particular Birdo as a species, it treats that one as a single character, and that particular character is said to think he is a female but in reality he is not.