Kratos: "It is a scorpion. It is in its nature to do harm"
Atreus: "That's what mom used to say about gods."
So she was fighting a "personal war" for survival, she was apparently one of the baddest mortals (maybe? who even knows at this point) around, she loved telling stories, she found her purpose in protecting the weak, and she fuuuuucking hated the gods.
Now that I put all this that I've learned together into one comment thinking about it a few minutes, I don't think I really needed this piece of information to see what she would have seen in Kratos. They were not very alike in personality, but it seems like they were very alike in terms of not just trauma, but in values derived from that trauma.
Kratos is not a man who lives to protect others. He is not a charming man. He is not a learned man of letters. But he is a man who was cruelly and capriciously betrayed by his gods, and whose resulting single-minded rage became the undoing of the gods themselves.
The first God of War game, the OG on PS2, was a classic Greek tragedy: Kratos, in divinely-gifted fury, slew his wife and daughter who weren't even supposed to be there; Ares had placed them there to induce Kratos to kill them, believing that doing so would turn him into the perfect remorseless death machine. This did not work, and Kratos dedicated himself to serving the other gods in hopes of atoning for his deeds in hopes of making the horrific recurring nightmares of his brutal slaying of his wife and child.
You, dear reader, and I, do not blame Kratos for killing his wife and daughter. We blame him for many things--genocide, warmongering and all associated cruelties--but not for Ares' cruel betrayal. The ancient Greeks didn't necessarily see tragic heroes through the same moral lens, though. A tragic hero in the Greek tradition usually wasn't at fault for the things that he would be held responsible for, while being responsible for far worse that they're praised for. Same goes here for Kratos. As a final labor of redemption, Kratos kills Ares. But Athena, who set Kratos on this quest, tells him after his long, difficult, painful journey filled with unspeakable violence, that while the gods absolve him of his sins, they can't make the nightmares stop.
Kratos views this as a final act of betrayal, and who could blame him? As a mortal man, with mortal values, what could he have possibly expected from divine absolution but to ease his suffering? Who cares if the gods forgive him, if he's still tormented by grief and shame? So he throws himself into the sea to finally put an end to his despair.
Naturally, Athena has other plans. She "rescues" him and he's appointed the new god of war, along with the immortality to never escape his suffering. The gods are capricious and self-serving, they can't even let the man die on his own terms.