See, when you make Charles feel trapped, powerless and horrible, he hits you in the face with - well honestly whatever he's got to hand, until you either stop moving or he can smack you off a cliff.
When you make Edwin feel that way he initially just tries to get the fuck away but then perseverates like an angry little wasp nest until he figures out exactly how to emotionally gut you and reverse the emotional power-positions.
This wasn't Edwin being nice. This is not Edwin reaching out in compassion to someone who is also suffering loneliness. This is him walking behind the throne and yanking aside the curtain to reveal that the Great Oz is just a fussy little man.
So bear with me: in Sandman, we have the scene of the duel with Lucifer in Hell, and some people kind of objected to the idea that "Hope" won that idea-duel, because it's not like there isn't anything that can kill Hope (after all, Despair is Dream's little sister and she does it all the time).
The thing is, Hope can also kill Despair. It basically depends which one of them is bigger at the moment. Which means there's nothing that can answer "Hope" that doesn't end up sounding like two kids going "yeah well I win infinity plus ONE!" "I win infinity plus TWO!!!"
And it's engaging in that, it's being the one who is left with no move but to be the first one to do that, that means you lose the duel. It's not about absolutes; it's about form, and performance. The one-upmanship is about style, form, how impressive you are to the audience, and the moment you start the cycle of "infinity plus ONE!!!" (or have no other moves but that), you lose.
This is actually a very similar moment. If Edwin had used any other attack on the Cat King (and let's be real: there are plenty of bad things to say about the guy that would be true), almost all of them the Cat King could riposte with something that would be equally damaging to Edwin, or he could deflect and neutralize.
The Cat King's entire schtick is Power. From the first minute he comes on-screen his entire performance is meant to blindside and overwhelm and portray himself as something not to be crossed. To some extent that's also why he has to "deal" with Edwin's use of a binding on the one cat: these are "his" cats, and if he lets this go by then he's essentially admitting he's insufficiently powerful to deal with it.
Powerplay is also the underlying factor in every other thing he does. Edwin's initial recoil and revulsion from the sexual proposition is a direct challenge/attack on the Cat King's self-perceived powers of attraction/seduction; so is the way he reacts and recoils in the little garden; and relatively brutally so in the moment of "this is all you are to me" in the woods. The problem is that as far as I can see he's in some ways just not that smart - he consistently misreads Edwin and what's going on with him (about the only thing he accurately perceives is that Edwin is both hugely in denial and craves being wanted; everything else he just kinda swings and misses badly), and then subsequently wildly misreads Esther's mood and capabilities, to the point of losing a life and potentially contributing to her getting the ultimate power in the area.
This approach may also be reasonably genuine in motivation (after all, among other things Niko is the only one in the show that actually speaks to him respectfully), but it's also still a power-play: he arranges to encounter Edwin alone, with an ostentatious mourning gift of a very magnificent specimen of the Mourning Flower (the lily), carefully staged to appear as maybe a better aimed version of what he thinks is appealing (note here that his attire is formal and elegant rather than his earlier chaos-brat vibe); even his condolences still contain a reference meant to remind everyone of his own status and power (not all of us have nine lives).
Edwin accepts the gift but also then proceeds to verbally gut him.
Here's the thing, guys: there is zero power in being identified as a lonely person. Being lonely is not cool. The Cat King absolutely wants to be cool and Edwin has just said You're lonely and the thing is that this is, in a battle of words and a dance of social bullshit, very much like Dream's answer of "Hope".
It's not that there's no reply as such, but there isn't really one that will make the Cat King look good. Especially since it's true: he is desperately lonely. He could manage a performance to wave it off, but he knows it's true, and there's nobody here to perform for, because apparently Edwin does too. And worst, Edwin isn't framing this as an attack - Edwin is being kind about it.
But he's also shredding all of the mystique, all of the powerplay, all of the more-than-human, all of the everything that the Cat King's been trying to establish, and replacing it with:
I see you: you're lonely and you're lying to yourself and everyone else about it with smoke and mirrors.
And then he says, we are alike: we're both lonely.
Thereby reducing the Cat King to no more than himself.
Even if you assume that the Cat King does have some kind of genuine romantic feeling here, every single step of the way he has been at pains to attempt to assert his position of superiority, from the bracelet-trap to the "I'm older and more experienced than you" to everything. The position in which he is happy approaching this is one where he is sweeping little Edwin off his feet in a haze of something or other, the magical Cat King ("I'm not a common being"), wise and worldly ("I like your secret parts") and powerful ("I'll stop playing nice").
And Edwin just went "yeah, nah, you're nothing more than me, and you know it."
This, by the by, is absolutely the kind of interpersonal shit that would have surrounded Edwin as he grew up; well-to-do Edwardian women were steeped in it, and frankly so even were gender-conforming Edwardian men, they just also sometimes punched each other. Peter Pan is contemporaneous with Edwin's adolescence and there's a scene in the novel in particular (because that's where you get the narrative voice which frankly Caroll is VICIOUS) where near the end Hook is desperately worried that despite it being tactically dumb as fuck, Peter is winning the fight morally because he was showing Good Form. That's this kind of playing field. (It's a parody of this kind of playing field, but it's still referencing the same forces of society.)
What this does is absolutely destroys any hope of interacting with Edwin on those terms that the Cat King clearly wanted (very clearly, from everything he does), ever. It's possible the Cat King has the sheer magical/whatever power to just smite-punish Edwin for that - this kind of play isn't necessarily safe - but he's stuck on the rails of Good Form if he does, the same way that Lucifer could have just Ended Dream for beating them, except that doing so would actually have been even more humiliating. It would be admitting that you cannot meet them on this field and also that they hurt you so much that you have to smash them to regain your own equilibrium.
Edwin has just told the Cat King that he has no mystery, he has no glamour: he is seen, and he is understood, and as a result he is reduced because his entire persona and presence in the world is actually meant to make sure that this is not what people see.
And Edwin knows exactly what he's doing, is doing it on purpose, and by doing it is taking back the sure footing that the Cat King shoved him off of in their first interaction. Which he can do now because he's not lying about himself to himself anymore; that's part of why it was so important for him to completely unburden himself of all his denials/etc on the long stairway even if it was a really, really stupid time to do it.
The Night Nurse yanked Charles into the horror that he tried so hard not to think about and it exploded in her face in a sense of yeah sure you can do that to me but you're not gonna like what happens when you do; yes, she hurt him, and yes, he then had to deal with being destabilized like that, but she didn't get anything she wanted out of it.
The Cat King yanked Edwin into inner turmoil about the shit that he'd just been shoved up in a box on his inner shelf because all of it hurt too much and he didn't want to deal with it, and bluntly he did it for his own amusement and his own gratification; it's a shitty thing to do and subjected Edwin to a lot of fear and distress, invading and even flat out coercing a lot of his private inner life out of him when he did not have any intention of going there . . .
. . . and this is Edwin magnificently managing a turnabout, ripping the last thing the Cat King would ever want anyone to see about him into the open between them, letting him know that not only does Edwin see it, but yes, seeing it has equalized them because now the Cat King is someone Edwin can pity. Can feel compassion for. He is not a powerful demi-god, he is not a magical mystery, he is not the older worldly lover, he is just . . .
. . . another lonely person. And Edwin feels sorry for him.
The peck on the cheek at exactly the right place to, from Edwin, balance between interest and the kind of kiss you give your aunt, is a fucking brilliant choice as well.
And, if you look at his expression and the steady look he has in the moment he says it (that first gif up there) Edwin knows exactly what he's doing and is extremely pleased to see how well it hits.
I fucking love him so much.
(I will give the Cat King this: he does manage to take the hit with dignity, and has the wisdom to withdraw from the moment and take the message of it. For now, at least.)