oh, anon. you've no idea the worm can you've just opened...
*ahem* Okay, primarily, ShuRai's appeal is actually one that's pretty common for me in a lot of my more beloved ships:
At their core, both Shuten and Raikou are deeply wounded children who were cast away and thrown aside for something they had no control over, their lineage. They are, the two of them, monsters. Literal, actual monsters, in a setting where that has tangible weight on both how the world perceives you and how you perceive yourself.
Both of them need to constantly grapple with what their half divine/demonic blood actually means, both of them need to reckon with how they view themselves as opposed to how the world views them.
The difference is how they dealt with it...or rather, how the both of them didn't deal with it.
Raikou buried her resentment and anger (personified as the emaciated and pale Ushi Gozen) as deep as she could possibly manage so she would be accepted by her family (especially her massive prick of a father Mitsunaka), outright refusing to cope with that and projecting that hatred of herself into her skills as a demon slayer. She surrounded herself with people who looked up to her, her generals, because she wanted to feel wanted and loved and so she projects that outward as well, her maternal feelings for Kintoki and Tsuna are genuine but are driven by a deep fear that she's less then human, that one day she'll snap and there would be no turning back.
Shuten on the other hand...gives up. Sort of. She doesn't engage in self reflection whatsoever, instead launching herself into drink and blood and indulgence because that's easier then confronting the tormoil inside of her. She outright considers herself a monster, admits it out loud, with pride, and with zero nuance. She's an oni, a demon, she kills people and eats people and there's no shame in it so long as she never ever ever thinks about it too deeply. Instead of actually confronting her conflicting inner nature she says, "I am Shuten Douji." And not as a statement of self-actualization, but as a statement of resignation. And as such Shuten refuses to be anything other then "herself" or rather the image of herself that she's built up over the years. She would rather die, literally, explicitly, then be anything other then "Shuten Douji." To consider if she ever had another choice or another path is literally impossible for her without just..snapping.
Both of them find their own sort of comfort, Shuten in her gang of Oni and Ibaraki, Raikou with her Kings and what praise she gets from the people she serves. But none of it addresses the root of their problems. For all they want to be one thing, human or oni, the fact remains that they are both, and this causes a rift and an emptiness within themselves that they just don't know how to heal.
And then...they meet. The demon slayer with hidden demonic blood and the demon with a touch of hidden human despair, and of course we all know how that went. Blood and pain and it ended with Shuten Douji dying...in more ways then one. It ends with Raikou looking down at the head of a demon that she killed by acting like a demon. And their issues die with them, unresolved as they fester in their graves.
And then...they meet again. In Chaldea. Alive again, with all those issues as ripe and raw as they were that fateful day on Mt. Ooe. And what do they do? They bicker, they needle, they fight and scream and annoy each other to no end because they know deep in their bones how similar they both are. And it pisses them off to no end.
Raikou and Shuten appeal to me because they are both broken people, broken in different ways but from the same source, alienation. Deep down all the two of them want is to not be alone. And since they are both of mixed blood and lineage, demon and divine and human, no matter how they meet...they will always feel that loneliness eating away at them.
Because nobody else gets it, nobody else knows exactly how they feel, not even themselves...except for each other. And that would just piss them off further. All that emotion and history and blood (both literal and metaphorical) between them is just ripe for boiling over into something explosive and violent.
Given enough time to adjust to each other and have to deal with each other in Chaldea on a regular basis, they would either get into a fight so violent it would rip the base in half....or have a night of fucking (not making love, not sex, fucking) so intense it would...rip the base in half.
Either way, they would finally get some catharsis, some outlet, some source of twisted comfort and relief from someone who actually gets it. And they could maybe, just maybe, find a way to help each other pick up the pieces after that. Maybe.