But the game itself implies that the game is a dream (and it was apparently one of those fake-wake-up dreams where you have a dream and then think you wake up and then you wake up again but for real).
And I'd also like to reiterate, if I may, that I think this whole argument is moot because there are undeniably multiple Birdos. There are, in all likelihood, at least three in SMB2, and at least one sports game clearly shows a whole crowd of them. Yes, the manual says "he" rather than "they", but it treats all enemies as characters. Look at the entry for Ostro again (ignoring the switched name):
It calls Ostro a he instead of a they, even though there are multiple places in the game where you see two or more Ostros next to each other. Blame the translators and the ambiguous language and non-American culture of Japan.
The characteristics attributed to Birdo in the manual probably don't apply to the whole species, however, because if the species has meaningful concepts of male and female, there must be at least a few actual females and at least a few males willing to mate with the actual females, because the species has clearly propagated. They can apply perhaps only to the first Birdo seen in SMB2, or to all of them in SMB2, or maybe Mario, for some Freudian reason, always dreams about Birdos being transvestites.
It is unlikely that the playable Birdo that's been showing up in sports and kart titles lately is the transvestite Birdo from the SMB2 manual. Even if SMB2 was real, it took place in Sub-Con, a dream world separate from the Mushroom Kingdom, and people probably don't permanently move from one to the other (Incidentally, another pet peeve of mine is people complaining about SMB2 enemies showing up in the non-dream Mushroom World. In-universe, it makes perfect sense that Mario would dream about some things that actually exist in his world.).