The root of the problem is that reputation, boundaries, privacy, propriety, etc. are all treated as very very important in Japan. But what they entail is also significantly different. Social awareness and maintaining a proper sense of distance are heavily emphasized, and ignoring or overstepping is a much bigger faux pas than in the West. Someone's problem(s) being their problem(s) is the norm, as is the idea that one should keep private matters private so as to not disturb the greater community ("community" being the classroom/school, workplace/business, so on and so forth).
So butting in on something that doesn't explicitly involve you is often looked upon very poorly. Luffy, Naruto, n' Ichigo, for example, are all treated as varying levels of clueless or reckless if not outright insane for getting involved in business that "has nothing to do with them" because the norm in Japan is to mind your own damn business. That is the context at the core of the MC's internal conflict when it comes to this current arc: he isn't sure if he has the right to meddle in the FMC's business when her behavior so far implies it's a private matter. He doesn't want to overstep and risk damaging their relationship by imposing himself on a situation where his involvement is neither welcome nor wanted.
To give a comparable example from Western culture, it's like... say a woman is being hit on by someone at a bar, then some dude who she barely knows comes up, wraps his arm right around her shoulder like she's his girlfriend, and starts asking "is there a problem here?". The woman did not ask for his help, the woman may not have needed or wanted his help (maybe she was handling it just fine, or maybe she was receptive to the person hitting on her), and he completely ignored personal and social boundaries by getting in her space and behaving in a way that is entirely inappropriate for nigh-strangers to act towards another person. All in all there's a significant chance the guy in this scenario is behaving even worse than whoever was hitting on the woman. That is the equivalent of what the MC fears he would be doing by getting involved without the FMC saying anything or asking for his help.
Meanwhile the FMC sees this as her personal business to deal with and not something she should burden others with. In Japan having to get others involved in something like this is generally seen as a personal failing, because you're imposing on and inconveniencing the people around you. Think of a drama llama; we've all known one. Someone who has to make any issue they're having, big or small, everyone else's problem. Or one of those big controversies between internet personalities (Vtubers, Let's Players, content creators in general) explodes, and by the time the dust has settled and all cards have been laid on the table, the prevailing sentiment ends up being "they really did not have to make this a public spectacle" or "they could have and should have handled this privately." That is the logic of the FMC and people like her in these situations.
Now: could all of this be resolved if the two just sat down and talked to each other? Yes. That very much is the case, and the manga itself is all but saying their lack of communication is making this situation even worse. But they are both dealing with a much higher hurdle than most people here realize, one born of cultural and societal expectations. They don't just need to set the record straight on everything that's happened, they also need to establish where exactly they stand with and what they mean to each other. But neither of them is taking that first step because they both fear overstepping in such a way that would upset or outright drive away the other. Because in Japan, doing so when it is neither wanted nor warranted is grounds for someone to disavow you entirely.