Not really, no. Bruce lists animals that display "transgender" traits that he categorizes into two subclasses as "intersexuality" and "transvestitism". There are relative few such examples given, and those animals have behaviors that are best described as sex/gender-non-conforming behavior, i.e. the animal is behaving in a way that is stereotypically associated with the other sex of that animal species. We of course know that this doesn't literally make someone the opposite sex, but rather that their behavior simply doesn't conform to sex stereotypes. Also, this is a book written in 2000 and terminology about this sort of thing was rapidly changing in the past few decades, so obviously the trans related definitions used then don't have the same definition as nowadays.

So no, this book can't really be used by the trans movement as a defense of transgenderism being naturally occurring in other animals in the same way it does in humans. There's nothing that really compares in other animal species with the phenomenon of transgenderism in humans.