So, for my personal portrayal of Hank, and how he sees Connor, it will be like a father and son relationship, which is just my personal interpretating for my Hank & Connor. There are two quotes that come to mind for the reason that I think that:
Hank: "Connor! Hang on son, hang on, hang on! We’re gonna save you, hang on!.. Here, here..."
This is a quote that's in response of Connor dying in his arms if you're not able to complete this level and put the biocomponent back into Connor that regulates the heartbeat. Hank comes running in a smidge too late to save him. I think there are others but I can't remember them right off hand.
Hank: "Every time you died and came back it reminded me of Cole."
I'm using this quote too, because even though he doesn't call him "son", he's comparing him to his actual son. And no, I don't believe that Hank sees Connor as Cole, but rather Hank has fatherly instincts still and Connor brings this trait out in him. You see Hank many times guarding Connor, asking him if he's alright, going into a room first with his weapon drawn and making Connor stay behind him, etc. He is protective over Connor despite him saying that he hates androids.
But it also shows that Hank is still dealing with the trauma of losing his son and he is sort of using Connor as an outlet to work through that trauma. I think Hank's Chief of Police saw how much Hank was struggling, and how much he had changed since he had lost his son, and knew that this may be an opportunity to sort of get Hank back on his feet again. His Chief knew exactly why Hank hated Androids, so he probably had this idea that maybe if he got to work closely enough with one, then maybe he would be able to work out his problems.
Hank didn't really hate Androids because he thought that an Android killed his son, rather he hated them because a human Doctor would have felt something, if his son died on their table. But an Android, at that time, felt nothing, which made Hank alone in his grief because they couldn't understand. They didn't care that his son had died on their operating table, but a human doctor would have and they probably would have tried everything to keep him alive.
Of course I don't think the Chief would have known how well his plan would have worked, because he didn't know that Connor himself would become a deviant... But that in itself is what snapped him out of his depression, because he saw that Androids were plenty capable of having emotions and caring. In fact, multiple times throughout the game, he pushed Connor into trying to feel and think for himself, just to see if he could, going as far as to point a gun at him at one point.
If you go down the route of "machine Connor", or you have Connor die one too many times, Hank will die in some form or fashion. Which I don't really want to go into because I feel like that belongs in a different meta... but suffice to say that Hank needs Connor to be alright, he needs to see him having human emotions and that's what helps him come to terms with the loss of his real son.
Connor doesn't replace Cole at all, and I'm not saying he does, but I feel like he sees a piece of him in Connor in a small way and so he treats Connor like a father would a son and through him Hank was able to see Androids from a different perspective than he would have other wise, and in that regard, Connor saves Hank's life.
And they also kinda look alike. I kinda headcanon that Hank sees Connor as how his son might have looked like if he were older.