I haven't actually researched this, but is "cross-dressing" in Japan another word for being trans, or is it really just a fascination with dressing as the opposite gender?
They're not really related elsewhere either. Cisgender homosexual or heterosexual crossdressing is not really related to gender dysphoria as much as wanting to look cute. That and not confirming to gender norm dress codes is unrelated to gender, there is nothing less legitimate about a trans female dressing as a male either.
I think something important to consider is cultural differences. I don't think transgender is really taken as seriously in Japan. Often times characters presented as transgender are described as crossdressing.
Like whatever uncomfortable miasma of gender identity "My Wife is a Man" lies in.
Maybe there is a cultural component to that but it's not like western cultures aren't guilty of conflating cisgender crossdressing with transness either. For a long time before research and awareness become more common it was all pretty much put under the same umbrella of "Homosexuality" (read: Deviance). Regardless it's pretty clear from the text here that the main character isn't trans.