Let me try to answer this. The answer to your question of is this a choice is, NO, it is not a choice. It is biologically-driven, but the phenotypes that arise are much more complex than simply external genitalia, and so in truth there is assigned gender, and there is true biological gender. What happens in the brain during development may or may not correlate to what grew in the groin area in utero. You can read my previous post on the genetic and biological basis of gender for more detail. For 99.7% of people, this is easy, coordinated and consistent. Assigned gender equals biological gender in these circumstances. For the other 0.3% of cases, gender assigned at birth does not correlate to the rest of their biology. The only choices to be made, then, are whether to live life as the wrong gender, or change to the correct one, and then WHEN to make any change.
So those of us who are men....think about all of our thoughts, urges, behavioral tendencies, etc. We don't really choose these. Some is taught by our upbringing, but some is not--it is part of our biology. It is encoded in us. Males and females are distinct, mentally as well as physically. This is controlled by brain chemistry, and brain chemistry is ultimately controlled by our hormonal axis. Now imagine having all of these same thoughts, behaviors, urges, etc, but being born with female external genitalia (or in some cases ambiguous genitalia that is guessed upon and surgically corrected at birth). Imagine the confusion this person must have. Almost all of transgender situations can be traced back to expression ambiguities or polymorphisms in either the hormonal axis genes or the sex chromosomes. So, no, in that sense you are correct--you cannot "choose" your gender, only whether or not be public as one or another gender.
To this end, the comparison to ethnicity is flawed. You cannot choose your ethnicity either, but there isn't necessarily an agreed upon "norm" to what ethnicity behaves as. True, if one is of a certain ancestry, one has to be aware of certain genetic tendencies toward some diseases. Yet this woman you speak of who "decided" to be African American cannot, in truth, choose this. What does that even mean? Any African American can do anything someone of European, Asian, or Hispanic ancestry can do, and vice versa. On the other hand, there is a pattern to what a man does, and a pattern to what a female does, and feels. In this, then, the bigotry here comes in if one does not accept a person for who they truly are, or discriminates against a person simply for who they are in an unwarranted fashion. We can debate the particulars, such as this great thread has done above, in that it would be more appropriate for Mack to wrestle in the male division. This statement is not bigoted. It also is not bigoted to be concerned about what exogenous hormonal therapy might mean for performance, as long as we keep it based on the science and not hyperbole. On the other hand, if we scorn transgender people, call them crazy, think of them as weirdos, or discriminate against them in an unwarranted fashion, then I would say that this would qualify as bigotry. So far the only example on this thread of that (someone referring to Mack as an "it") was taken down. I hope, as a society, all of us can have the type of discourse shown on this thread. If so, we can help transgender people achieve normalcy in their lives, and be better for it.