Just thoughts on DiaDop. These are just my ponderings. Not meant to be in canon or not. Just exploring the character for my process. It’s just super rough and rambly.
Diavolo is constantly striving to maintain control. This thin veil that is merely a semblances control weighs heavy on his mind. He functions from a place of scarcity. Constantly striving for more- more everything because anything less is not enough. The fear that everything he’s worked so hard for can be torn from him in an instant is what drives him. He takes little joy in life, let alone himself. His fear and anxiety is all consuming.
The only way he indulges in anything is through Doppio. He uses this man as a surrogate for any positive emotions. Emotions that he can not afford. Emotions and experiences that, as he deems, would make him a lesser Don. Emotions that would threaten his all too fragile empire. Unable to distance himself enough from his own insecurities, he lives totally and vicariously through his Consiglieri.
The only way Diavolo shows himself love is through his precious Doppio. Diavolo keeps himself pent up, consumed in the stress of his empire, but he allows Doppio to go out into the sunshine and enjoy the little things in life, like a cute little frog or bug. Diavolo allows himself to live life and more importantly, allows himself to enjoy life through Doppio. Diavolo can’t afford such measly pleasures, himself. [In this guise he can allow himself to do good for others too.]
Diavolo even extends this further by asking Doppio about his day. Giving Doppio the encouragement to have fun, enjoy food, or even enjoy life is giving himself the permission to live vicariously through his Consiglieri.