There's a couple plural/singular possessive confusions, but I didn't notice anything egregious.
BLP
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To answer your question: Yes, there is something wrong with calling someone "black male" (also "white female" (also "female")).
Describing someone with a color/race-gender construction is generally only acceptable in technical documents like medical reports, and even then it's typically supplied as contextual information before moving on to referring to the subject in more specific terms (such as their name). Sometimes it comes up in a news media context, but typically (again) to provide context or when the subject's name is unknown.
When one uses a race-gender construction in standard American dialect it is...awkward at best. It's depersonalizing and clinical. It's jarring when Caelen refers to people he says he likes as "the black male" or "the white female". Generally one refers to people they like by their names. When not, one usually knows enough about them to be able to refer to them by traits more specific or relevant than their color or gender.
Writing is hard, and there are only so many ways to describe recurring characters, but avoiding tonal dissonance like this is important unless you're trying to make it seem like the speaker is telling on themself without explicitly telling the reader so (and I don't get the impression that Kael is intending to tell the reader that Caelen is conflicted about or actually dislikes Vulgor or Verissa).
Beyond the purely tonal problem with using a race-gender construction in this context, it...uh...how to say delicately? It's sort of like when someone's 400 year old grandma asks you to pass the Brazil nuts, but she doesn't call them Brazil nuts. Or...like when somebody calls a woman a "female" and you have to play the "is this person an incel or just socially awkward?" game. It's super uncomfortable and not an especially safe situation to be put in.
Though at the end of the day Kael isn't American and I really don't have enough experience with any of the dialects of English spoken in the UK to be able to say with confidence what would sound awkward, nevermind what sounds low-key racist or sexist. All that said, pretty sure "don't say 'feeeeeemaaaales'; it makes you sound like a Ferengi" is a meme so old it made it to Facebook a few years ago so...at least on that one, he's not got much of an excuse because we all know he's got an Internet connection.
I'll preface this by stating that I like this VN enough to keep reading it.
But I gotta admit that the OP makes sense.
I'm also confused why the perspective character insists that Vulgor is a good person. His poor impulse control paired with his tendency for violence toward those in his care aren't great signs for his, you know, moral fiber.
This is going to seem like an aside, but bear with me, it's going somewhere: I don't know if this is a culture issue, or a dialect difference, or what, but using "female" instead of "woman" and "the black male" instead of almost any other description are pretty cringe-y word choices. Couple Vulgor's characterization (short temper, sexist, quick to violence) together with the narrator's favorite description for him (the black male) and you get one whole yikes (that's the proper lit crit term, btw). I don't think that knot is intentional--but it's still real uncomfortable for me as a reader and contributes to the impression that the racism stuff that's actually intentional isn't being deployed with appropriate care.
So, have I enjoyed FBTW? Yes. Does it make me cringe sometimes? Oh, yes.