Hope answers from the light novels are okay. I remember the anime following the LN very closely on this point, so it shouldn't affect the outcome.
Black Hanekawa's biological sex is female. Its body is the body of Hanekawa Tsubasa, who is most surely female. (If anyone has any doubt on this point—look at what Koyomi's always fixating on.)
Oshino does some speculating in Neko Black, but the most definite information comes from the later parts of Nekomonogatari White. The issue of its gender identity is far murkier, as it tells us itself in Nekomonogatari White, Chapter 61, just after it reads Tsubasa's letter on the origins of Kako, in which she calls Black Hanekawa her little sister:
Besides, I was male when when I was run over, so it's weird to call me a little sister—and in the first place, even though my base was the Hindering Cat, seeing as I was created by raw material cut off from Mistress, my gender isn't just vague, it's undeniably a grey area between being a sister and being a brother nyah.
What's the point of asking an abnormality about its gender, anyway?
Here, Black Hanekawa calls into question the idea that kaii even have genders or that the concept is meaningful for them. (After all, they don't seem to use sexual reproduction.) Even Shinobu, who is obviously female, seems to be pretty much the same as male vampires except for outward appearance; her sex is a holdover from her time as a human, when it did matter. (See Kizumonogatari on Shinobu once being human.)
Given the context of the above quote, it seems like Black Hanekawa was trying to mock Tsubasa for getting things wrong (against her usual inclination). Just before that remark, it says
That said, it looks like Mistress has still got a few misunderstandings nyah—though I suppose, seeing as she was conjecturing with a lack of data to base her decisions on, these mistakes couldn't have been avoided nyah.
Plus, both the style and content of the letter [that Tsubasa wrote to explain the situation to Black Hanekawa] are all over the place by Mistress's standards—it definitely wasn't something written while she was calm nyah.
It isn't a situation to be hoping for full marks anyway, so getting an 80 for an A-grade is good enough nyah.
So the above quote is in the spirit of continuing to call out Tsubasa on her minor mistakes—she calls Black Hanekawa her little sister, and it responds by nitpicking her for assuming it was female. That suggests this isn't a serious statement about Black Hanekawa's gender identity.
In Nekomongatari White, Chapter 60, Tsubasa speculates that Black Hanekawa existed inside her long before she ever encountered the Sawari Neko. In Neko Black, Oshino hypothesized something similar—that the Sawari Neko's consciousness is completely gone from the equation and it's really just its powers being directed by Tsubasa Hanekawa's unconscious will that comprise Black Hanekawa. If true, that makes the Sawari Neko's gender even more irrelevant. But none of this necessarily makes Black Hanekawa's gender identity female—it could be a suppressed masculine part of Tsubasa that expressed itself in the fusion of Sawari Neko and Hanekawa that created Black Hanekawa.
Kako, the Inflaming Tiger,
is also a spinoff of Tsubasa, yet is always treated as male, except when Tsubasa calls it her "little sister".
But there's no evidence I can remember in the books or anime to say that Black Hanekawa actually identifies as male. It's murky, for lots of reasons, and I think it was presumptuous of the wikia to post that line as it did. Furthermore, it seems pretty far out there to assume that this means Tsubasa herself has a nonbinary gender identity, whatever the situation with Black Hanekawa is. Tsubasa is in love with a male, Koyomi. The imagery in the anime during Tsubasa Cat Part 5 even suggests she wants to have a sexual relationship with him. Tsubasa seems uncomfortable with Hitagi's grabbiness earlier in Neko White when they shower together. Nothing she ever says or does suggests that Tsubasa Hanekawa doesn't identify as female. The most we could say on that score is that there's a vague possibility she may have had something like a split personality inside her that identifies as male—and that's almost pure speculation.
Sorry to burst anyone's bubble, but the evidence does not support the idea that Tsubasa Hanekawa is not biologically and psychologically female. And the evidence for Black Hanekawa is very ambiguous, and muddied by the irrelevance of gender for non-human kaii.