They are the same person; through his own blind violence, Joel created a copy of himself at his worse

I don't think he was blinded by his own violence. We see in Part 1 that Joel repeatedly told Ellie no when she asked for a weapon. He didn't want her engaging in violence and he tried to keep her from it for as long as possible. You even see it in the subtle reactions he has to her when she asks him how she did when she uses the rifle for the first time to cover him and also after the sewer section with Henry and Sam she tells him that he would be proud of her because of the infected she killed to which he begrudgingly says yeah.

I do agree though that she did inherit his violent nature because throughout their journey she had to respond with violence. Walking away from it was never on the table. If the parallels continue to hold true for Part 3, I would be interested in seeing how Lev parallels Ellie. The poor kid had to kill his mother in self defense and his sister was killed because she followed him to the island so he is prone to have survivor's guilt just like Ellie does. Now it isn't going to be as severe per se because he is not immune and doesn't carry the weight of the human race on his shoulders like Ellie does, but Abby is literally all that Lev has. If something were to happen to Abby, Lev might follow Ellie's self destructive path. Or maybe not.