edit (sent the answer too fast)

and they insist that you don’t feel the way you are feeling

okay, I’d say that that is another problem than not wanting for apologizing for one’s actions.

It’s that I feel a certain way, and they want me to feel differently without apologizing for it.

I don’t think that most people want the other person to feel differently but to make them understand that they didn’t hurt them on purpose (that’s why they empathize the ‘but I didn’t mean to’) but then they get defensive when they are asked to apologize because to them that means that the offended person thinks that they did do it on purpose, that they did it to spite/hurt them. Like unknowingly bringing up a sensitive topic during a conversation. They don’t want to apologize that they spoke about this topic because they don’t think that it’s a inherently ‘wrong’ topic to talk about. They do, however, want to convey that they didn’t know that this is a sensitive topic to you and they wouldn’t have brought it up if they’d known better. At least that’s the way I see it if I’m in the position of a neutral observer in this kind of situation.

If your counterpart is actively trying to manipulating you that’s different of course but I didn’t interpret the comic from that point of view.