What do you mean by "offensive" and "objective"? The thing is, of course language can be used offensively to attack people. For example, humans are social animals, so society and the standing in society is crucial for us; and so if you shame, humiliate, outcast someone, diminish their social standing, spread lies, or otherwise attack their social standing then in previous times that could have meant death. Like for real; if your herd expels you, you'd have to do everything on your own, and more often than not, the individual is not capable of doing that and, even if they are it's not an appealing idea to many.
So things like that might be existential threats to their existence, and in turn might prompt ferocious or even violent reactions. Like seriously; a divide between "us" and "them" is listed as the first stage of a genocide:
https://en.wikipedia.org/wiki/Ten_stages_of_genocide
Also, the information that you have shapes how you feel about a situation. So if someone tells you that you suck and you don't have sufficient support that makes this an outlier then you have a high likelihood of having that affecting you. You might crush someone's self-esteem or give them a feeling of unease, of not being welcomed maybe even being in danger.
So yes words CAN very well be used offensively in order to attack people, and they DO tangible harm. Science has evolved a lot since sticks and stones and I'd be surprised if you find serious researchers that argue that non-physical interaction can't do harm... Like seriously, even doing nothing and ignoring a person can mess with us.
That being said, just because an action is perceived as offensive doesn't mean it was intended as such. Like IDK, if someone is first primed with pictures and descriptions of Hitler and is then shown Charlie Chaplin and shows a reaction of unease because of the mustache then that is not necessarily Chaplin's fault. It might just be a misunderstanding, an unfortunate coincidence (and it's even Chaplin who wore it first).
However, despite not being intended, the reaction is probably still real. Like you might chuckle once you become aware that it's just the facial hair that makes them look similar, but for a split second you might have an averse reaction triggered by such a benign similarity.
And that reaction is a fact, it's objective reality, even though it is a subjective reaction, and the same word in different contexts and to different people might trigger different reactions.
Where "trigger" here means that a seemingly small action "like pulling a trigger" causes a much more violent reaction "like what happens after you pulled the trigger (of a gun for example)".
So if it is a genuine reaction it's pretty valid; what are you supposed to do about your feelings - you can't change them in the moment, can you? And the more you are forced into an emotionally disturbed state, the less likely it is to stay or get calm again.
So on the one hand it's less about the specific words themselves but about interpersonal relations and assumptions that are used to attack a person or that are falsely assumed to be attacks. But again those might not be related to specific words, like you can insult a friend all you want and they might not bat an eye if you generally don't give off a vibe of hostility while doing so, but a seemingly benign word that makes them question the seriousness of your friendship might be a lot more hurtful than any insult.
On the other hand, it could very well be very specific words, symbols and slogans that are offensive, as those can very much be used to convey a sense of "in-group" and "out-group" as well as providing a shorthand for an entire narrative. Again being told you're not part of the peer-group or being ridiculed and outcast for who you immutably are is an act of aggression or at least is an active threat to a person's wellbeing. And common language and symbols is a mean to define who's a member and who is not.
Though these words and symbols are not themselves objectively offensive, in fact they might be totally arbitrary - it's about the context that they create and reflect that is the problem. So even though they might appear to be seemingly objective, that may change over time and context. Like if in medieval times when the sick were left untreated out of town, you'd had a massive aversion against people calling you sick and it would be an instant insult because the consequences of that being true would be abhorrent to you, however if you have decent healthcare and there's no social stigma involved there's no problem to declare yourself sick and take a few days off to fix that.
So with regards to your example, person A might not initially tried to attack person B, but the total disregard for person B's reaction and the lack of trying to clear up this apparent misunderstanding still makes them look like an asshole.
