"What is REAL?" asked the Rabbit one day, when they were lying side by side near the nursery fender, before Nana came to tidy the room. "Does it mean having things that buzz inside you and a stick-out handle?"
"Real isn't how you are made," said the Skin Horse. "It's a thing that happens to you. When a child loves you for a long, long time, not just to play with, but REALLY loves you, then you become Real."
"Does it hurt?" asked the Rabbit.
"Sometimes," said the Skin Horse, for he was always truthful. "When you are Real you don't mind being hurt."
"Does it happen all at once, like being wound up," he asked, "or bit by bit?"
"It doesn't happen all at once," said the Skin Horse. "You become. It takes a long time. That's why it doesn't happen often to people who break easily, or have sharp edges, or who have to be carefully kept. Generally, by the time you are Real, most of your hair has been loved off, and your eyes drop out and you get loose in the joints and very shabby. But these things don't matter at all, because once you are Real you can't be ugly, except to people who don't understand."
"He longed to become Real, to know what it felt like; and yet the idea of growing shabby and losing his eyes and whiskers was rather sad. He wished that he could become it without these."
After getting to the end of TLOU2 the significance starts to make sense.
Becoming 'real', or feeling complete, is the course a life takes and the decisions a person feels they have to make. Love can also be painful, because it opens one up to potential loss, and actions taken out of love are not always rational.
Ellie was incomplete without Joel, and on top of that she'd failed to forgive him in time. Prior to this she'd lived with Joel's lie that her immunity had meant nothing.
In the final flashback with Joel she says, "I was supposed to die in that hospital. My life would have ****ing mattered. But you took that from me."
Joel's response is, "If somehow the Lord gave me a second chance at that moment...I would do it all over again."
No matter how much pain it caused, he couldn't have let her die. She mattered that much to him.
But Ellie responds, "I don't think I can ever forgive you for that...But I would like to try." To which Joel replies, "I'd like that."
This is the literal hurt that Joel goes through for loving Ellie as a daughter. No matter how much his decision hurt her, nor how much her lack of forgiveness hurts him, he will suffer the pain and never stop loving her. As the Skin Horse said to the Rabbit, "...these things don't matter at all, because once you are Real you can't be ugly, except to people who don't understand."
At that time Ellie didn't understand, but she "...would like to try."
Hunting Abby is the thing she does out of love for Joel, and the necessity of finding some sort of purpose and completion. It's the act she can accomplish that will finally prove her forgiveness.
The same can be said for Abby. Joel had taken her father from her, and she needed to find completion.
However, they were both misguided. Revenge wasn't the way to honour the dead, and it broke both of them. Like the shabby, worn out Velveteen Rabbit, Abby was a shadow of her former self, and Ellie was minus two fingers. Rather than finding completion Ellie found more loss, and failed to transform into the 'real' rabbit. If she'd stayed with Dina on the farm she would have had that chance.
At the beginning of the game as soon as the player takes control of Ellie, there's a rabbit outside the window. It may be a foreshadowing of the 'real' rabbit she wants become, when she has forgiven Joel and found completion: