Pretty sure it’s part of the plot. The old vet lady obviously knows. The brother too also obviously knows but chooses to call him his “sister” he also said their dad would complain about his hair being so long. So I’m guessing he wants to live as a woman or something? Also I think in chapter 17 or so, when he was changing you can see a picture of him with another guy. It looks like he was wearing lipstick and makeup in that picture. So my guess is he is either trans like you said, a drag, or he’s gay and his ex made him dress like a woman. Idk but I think it’s part of the plot.
Additionally I hear in Japanese they don’t really specify gender in the language. So I could say “he’s my boyfriend” but in the language it’s not specific, it could be a him/her I could be talking about. People just go off of context.
Considering he was focused on Shizuma not hiding his relationship, my guess is the director is gay and started passing himself off as a woman so he could publicly be out with a boyfriend. Though I assume that relationship ended poorly, probably because of his job as a vet.
The gender language thing is actually kind of the opposite. While I don’t think there are strict male/female pronouns there are multiple versions of thing like I or you and they connate masculine/feminine as well as levels of formality/respect. A “neutral” term would be a polite term used by women generally or men when they are speaking to a superior (or similar situation). A casual example would be that -kun could be used for a girl but it’s rare since is it has a more masculine implication and how -Chan is sometimes used for young boys or feminine men.
How come everyone thinks he's a woman though? Do the other employees really not know that he's man or are they just pretending? Considering what we've seen of his character so far it doesn't seem like he would care enough to correct people who believe he's woman, but still how does that work for people who see him everyday doesn't seem like he's actively trying to hide it