It's not for lack of memory that he won't have mentioned Lynette's surname, it's simply not been necessary. I'd say it's far more likely for Neil to have said something along the lines of, "Your mother's name was Lynette", than to say, "You mother's name was Lynette [surname]". Even for the MC, to ask after the first statement, "what was her last name?" does not seem like a natural follow-up. It's irrelevant information to a child who may also naturally assume that their parents were married and therefore shared the same last name.
I've known people who don't care to know much about estranged family members. An ex-gf was adopted and I asked her once about her real parents and she said she "couldn't give a flying fuck" who they were or where they were, and a friend who's Dad left when her Mom was pregnant said she had no interest to meet him or know who he was. Granted, the death of a parent is somewhat different, but the MC's been told by Neil about how Lynette's family disapproved of him and pretty much shunned her for running away with him, so I can see how, because of that, the MC doesn't care to know who these people are and therefore to know what her surname is.
This game is not The Blacklist. It's one of the furthest things that DPC could possibly gleam any inspiration for this game from. Stop using it as a reference point for theories.
Protagonists and antagonists having romantic or even just sexual relationships is not something new or unheard of. Batman fucks Catwoman on the regular and she's been considered an antagonist for most of her existence. Another most famous example is Gone with the Wind where Scarlett O'Hara (the protagonist) has a relationship with and even marries the antagonist Rhett Butler.
An antagonist is not always the same as a villain, they're just the character who pushes back against the protagonist, blocking them from certain things, and proving to be a veritable thorn in the protag's side. The antagonist's efforts to reach their goal can often determine the story’s path and the escalating conflict, and this description seems to fit Quinn perfectly.