If sexual behavior does not equate to sexual identity then how one identifies is meaningless because the decisions and actions of the individual is paramount. I can identify as a parrot, but if I don't have brightly colored feathers and fly around the jungles or rain forests of Australia, Asia, South America or Africa, then I'm hardly a parrot.
No, it does not work that way. Terminology has been created by man to succinctly identify ideas, objects, people, etc. However, terminology is not all-encompassing.
With my previous reference to men who were 100% homosexual, but had sex with women, for example, they had sex with women out of sense of survival, on some level. They did not want to risk losing their lives, livelihoods, relationships, etc. They were (or are) still completely gay, but in terms of behavior, they had sex with women to conform, thrive, and/or survive.
The analogy using the Parrot falls short. It would have been better had you been more specific and said that you can identify as one kind of Parrot with a certain set of features, life expectancy, regional locality, etc. However, you identify as another kind of Parrot with different aforementioned traits or characteristics.
While a Parrot is a Parrot, there are several variations in the species of Parrots. The same applies to humans. There are several variations for human sexuality. However, a human is still a human.