Before, he says "俺は自生の効く男", which to me sounds very much like he's calling himself a guy. And even so this is Jinguuji, I don't see how would he not use "he" when talking to Tachibana. I forgot that in this very chapter, when thinking to himself, Tachibana seems like to currently think of himself as a man too. All pieces in the context points that "they" is simply weird to be used by them.
Tachibana is referring to himself as male here because of how he thinks, which is reflected in how he freaks out at Jinguuji saying that he's a girl now; he's thinking simply, in terms of physical body (also used because he's uncertain of himself and thus finds refuge in the physical reality rather than examining things deeper), while Jinguuji tends to refer to him not just as a him, as in those lines provided. It's a difference of who is thinking and speaking, primarily; I do agree with you that "they" comes across as weird because it lacks the cultural context and has its own burden of context in English, these days, but it is more accurate than localizing Jinguuji's words and thoughts of Tachibana to be gendered.
Isn't even the whole point of the story to be "funny story about a weird love between two middle aged guys"? If they stopped treating each other like that, it defeats the point and it defeats the parts reinforcing this gap. When they think of the other dude as a female, it creates a contrast exactly because in the inside they are both man. If the goal was to add ambiguity at all parts, I imagine their speech would change too. This is not done because it would hurt the romance and humor.
No, it's "Two middle-aged businessmen who are really close friends, one serious and the other goofy (basically a manzai duo), get isekai'd and one gets genderbent and given the charm of the Goddess of Love, and wacky hijinks ensue from there as they need to deal with their new circumstances". Tachibana has been slowly but surely becoming uncertain if he wants to be a guy anymore and settling into his female self while becoming much more keenly aware of Jinguuji as a member of the opposite sex; it's only the current storyline that inverts things and gives him time to re-examine himself as male versus Jinguuji being female, yet still very much set in his identity.
The tricky part is that in Japanese, using "自分" in that phrase sounds natural, not like if Jinguuji was picking his words to not "misgender" his friend or because he is simply unsure. Don't they also banter to each other based on the sex change, like in the image I posted?
Mhm. That's why I pointed out that it's an issue of Japanese being translated to English, with there also being a bit of a bias by the team translating that makes it feel a bit suspicious, but is in fact supported by the literal words and their meanings that are used; I will always side with the word and intent of the author(s) for how something should be interpreted over anything else, and where as here they explicitly use ungendered pronouns, though it reads weirdly in English because we have become unused to "they" as a singular pronoun over the past century-ish, I'd rather as close as possible to the same intent and framing be used.
When I pick any chapter at random and read their talks, I absolutely can't see anything that points for the use of "them/they" by either. If everyone used that kind of reasoning for translating every manga, translations would become unreadable. "Well, the character is referring to a girl, but without the usage of "彼女" or something that makes the character sex very clear, therefore, to avoid localizing, we should not use "she" or "her", it would be wrong since it is not written in Japanese like that.
This isn't a, "they don't explicitly use a female pronoun, ungender it" situation like you're presenting, this is a, "they explicitly used an ungendered pronoun, don't localize it to a specific gender" situation. Just like how I can say, "Oh, they're doing well" when asked how a mutual friend is to which both myself and the person asking knows their gender (this is also common misdirection play used in Japanese when the gender of a character is hidden for a later reveal, often as a gag). If Tachibana uses a male pronoun, it should be used, same with Jinguuji- however, often it's indeterminate and thus wrong to do so, even if MTL shoves a gendered pronoun into it (an easy way to point out an unedited MTL translation is wrong gender pronouns being used to refer to someone, because of this).
Lets use the prior quote image's dialogue/banter as example: [が 落ち着くし からだ 身体が変わった以外は 最高の気分だが?] [クソッ!! どうして おまえには魅了 かかんねえんだよ! 俺は今こんなにも イケメンなのに!!] [自分で言うな 自分で] This banter can all be loosely MTL'd as, "I feel calm, and other than my body changing, I feel great." "Shit!! Why can't she charm you? I think he's so handsome right now!!" "Don't say it yourself" and that's mostly correct, except for, "Why can't she charm you?"; That more accurately should be translated as something like, "Why can't I help but be charmed by you?!" (which I'm guessing the translated page does, or close to; I can't remember which chapter that is and since you didn't source it I ain't scouring the chapters here for it).
In here, for example, "あいつ" is used and no sex specific words are used, therefore to avoid localizing, we should not use "she" or "her"?
Correct, if that's all the context you have, as Japanese is a highly contextual language. [あいつ] "Aitsu" has two main uses; first, as a third person personal pronoun, used disrespectfully or with a sense of familiarity. Secondly, as a demonstrative pronoun, used somewhat loosely to refer to someone that they're emotionally close to, or someone that the speaker/listener already knows. It is used both by men towards other men, and by women towards men (esp. that they're romantically attracted to), but, again, context needs to be there in the first place; it is a secondary characteristic, not a primary one. "that one" or "that guy" or "that person" are all fairly accurate, but "that guy" (or even "that dude", if the character thinking/speaking would use such terminology- and that's walking a thin line, since in many cases translators/localizers choosing such is adding a personality trait or traits that was/were not there originally) would be the most accurate for English for both the informality and the intimacy concepts. In short, it's just like the above [こいつ] "koitsu" that I already addressed.
Remember that indirectness in Japanese is considered as being polite, and being direct is more for informal (especially intimate) situations; that's why most love story manga make such a big deal about not using last names, or not using honorifics, between people that begin dating, and how it's seen as shameless if they do that sort of thing out in public instead of when it's just a private situation between them, and so on. Being informal could be a sign of a rough and tumble gangster, or it could be a sign of intimacy, of a close friendship or a relationship between two people.