As Pucci has said, he loves Dio as he loves God. For a Christian, my Love for God is more like my love for my parents, indebted, but you wish to repay the debt. You see the debt not as one, but a chance to show your God that you're not a freeloader.
Dio changed Pucci's life, showed him a path through a dark hour in his life.
Now Dio's side of things is bizarre, Pucci was not outright mean to him, but neither was Jonathan, Dio had also gone on to say that for his Made In Heaven plan he sought a person who was not swayed by the delusions of power or sex, one who would be loyal and followed in faith with God. For indeed his quest was to get into heaven no matter the cost. Though it is clear that through time, Dio formed an interest beyond simply using Pucci, and it was clear by the bed scene, which was rather odd he was either testing Pucci for his plan, or testing Pucci for himself.
In the end, all humans (even those who reject humanity) want a place, want a people to belong to. Want to talk, even if they're supposed stoics or introverts, they're still projecting an image they wish to be seen as. They still wish to be seen.
Is it then possible to believe that Dio had been moved by Jonathan's death? That after the madness of youth, his want for heaven was a fear of damnation for his own grotesque sins? Was it a stubbornness to not go where is father was bound, to completely surpass him? Whichever it was, his nature changed slightly, and being with Pucci calmed him, he was with one of faith, one who did not judge him for his sins, one who was helping him reach salvation. As much as he was saving himself, Pucci was saving him as well.
What do I say? Bizarrely Platonic.