It's also not circular reasoning any more than it's circular reasoning to define Republican as someone who identifies with the Republican party
Nope. Putting aside the fact that “Republican” technically has an extremely objective definition (someone who is registered the Republican Party), if someone asked me what the Republican Party was, I could respond with “The Republican Party is an American political party which tends to promote policies such as…” In other words, I could tell you what the Republican Party is. Meanwhile, if I you told me, “a woman is someone who identifies as a woman,” then the term woman becomes so meaningless one would wonder why it even exists. At that point, we might as well just refer to sex, but doing so would be considered “transphobic.”