I mean, she is the "godmother". But it is clear that she treats MC totally different from the beginning. Why, we don't know. But the rest of the soldados do not live in the main house, spend time with the Deluca princesses, eat with them and gets to the point that even MC wonders if Cordia could be his real mother, after his real mother has treated him as garbage his whole life.
Probably a lot of overlapping feelings would come into play in terms of Cordia's feelings towards the MC.
- If Cordia and MC's father were ex-lovers (before MC's father was with Rina), then it could be the fact that she sees a view of MC's father when he was young (since even "Serpent" says to MC when they first meet "you look like your father).
Cordia's current feelings towards MC's father is not known, but it could be a simultaneous love & hate relationship.
But even should she hate MC's father, Cordia knows that MC is not at fault for any sins of his father, so perhaps that's why she treats MC different from others, despite under contract.
- Also MC's inherited a lot of his mother's facial features as well, so on the other hand, it could also be Cordia's continuous friendship towards Rina even to this day (though I'm not sure Rina feels the same way. It could possibly be that she hates Cordia) that comes into play as well.
- As you said, Rina really didn't treat MC well growing up, and with Cordia keeping tabs on Rina and MC's status, her compassionate side really came into effect, seeing MC looking down in her office nightie event (she showed some of this side to Isabel as well, when she first came into the Family, as Isabel reveals in her event with MC)
- But perhaps the biggest reason would be MC's personality and attitude towards life. The first question she asks him in her office in the introduction was "What is your ambition?" and MC's answer was (if I'm remembering it correctly) "To be happy."
It wasn't any loft ambition such as "I want to be rich and famous." or "I want to be powerful." the pitfall that his father has (as mentioned by "Serpent", MC's father's ambition basically burnt anyone who got in the way with his ambition or something along those lines).
Probably had MC said something like "My ambition is to be rich." He would have been sent to live with the Soldados like Cordia and Wilfred originally planned. But that all changed with MC's I guess somewhat "innocent" answer, which brought a slight smile to Cordia with her response "That's a good ambition."
Also, following is just my interpretation of it, but with Cordia being impressed with MC's answer to her question in their introduction, I think what she perhaps desired by having MC there as part of the Family is to have a semblance of normalcy living "Happy" lives as MC's ambition; not just for the girls (who each have their own issues, which MC begins to solve one by one, but for Cordia herself as well (perhaps she wants that innocence to rub off on the Family, even in a Family whose daily lives involve acts of violence among others).
I mean, yes she is plagued by guilt (as Wilfred says to MC), for things that the Family went through but sometimes one desires something even without realizing it themselves.