I found out recently that some people don't like Thresh and this upset me. Because I think he's great. I consider him the Patron Saint of "Talk Shit, Get Hit."
Futher, I think there are a couple times in the text when Collins intentionally links Thresh and Katniss.
"If only I was his size, I could get away with sullen and hostile and it would be just fine!" - THG
Basically, he is who Katniss wants to be. And from what little we know of his home life, that is not dissimilar from hers either. The family that represents him in CF is a younger sister and a grandmother. A little sis and a sickly parent figure. Same as Katniss.
Also in THG, Katniss apologizes to Lavinia saying she wished she intervened when she was being hurt. "It was wrong," she says, very clearly, to sit and watch while someone is hurt. And Thresh is someone who does intervene when he sees something he doesn't approve of happening. He rescues Katniss.
Thresh must have been hidden, as Katniss and Gale were under their rock, but he chose to come out and face the danger.
“You said her name. I heard you. You kill her?” Another thought brings a fresh wave of rage to his features. “You cut her up like you were going to cut up this girl here?”
And while it is the mention of Rue that spurs him to action via anger, not Katniss's plight necessarily, he does disapprove of Clove's actions. And I don't think anyone would blame him if he killed Katniss and didn't ask her this,
“What’d she mean? About Rue being your ally?”
He's curious about what happened to Rue. I always assumed this was because he felt guilty he couldn't help her. (Katniss goes on about how Gale, Cato and heavy people can't climb trees. Thresh is the biggest. There's real reasons he can't be with Rue besides the fact that a 12yo has never won and she has to die for him to go back to his family.) So when Katniss speaks of singing her to sleep, he feels he is the one failed to help someone.
Conflicting emotions cross Thresh’s face. He lowers the
rock and points at me, almost accusingly. “Just this one
time, I let you go. For the little girl. You and me, we’re
even then. No more owed. You understand?”
And obviously, the most obvious reference to Thresh being similar to Katniss is both of them being concerned about "owing."
“I think we would like Thresh. I think he’d be our friend back in District Twelve,” I say.
It's interesting she says "our" friend and not "my" friend. She doesn't think he's like Gale, her friend. It's not that Katniss considers herself to have a lot of friends. He's someone both Peeta and Katniss would like and get along with, which I think is a higher bar.
