What makes them unique? Just that they're in a game with good writing? Seems like the standard father-daughter story with an adopted daughter. But I don't know too much about the game, so... elaborate, please?
I don't see them as "unique" in the sense that almost all characters share traits. What made them work, and be believable, was the combination of consistent characterization shown through both their actions and how they act (this includes written dialogue but also behavior)
Take Joel - sure he could easily be a cliche but right from the first moment he appears he behaves right for his character within context. In the opening we see both his overriding concern and care for his daughter, his ability to fight to survive and the selfish streak of the true survivor in his actions towards others. When we pick up the story later he is subdued, too much the survivor not to keep living - and to an extent excelling at it to an extent - and with a brutal side to him that makes sense in context to his past and current circumstances. The first "mission" as well as being a classic tutorial also makes it clear you do not want to be between Joel and information or something he wants. His warming to Ellie is gradual and makes sense, reaching its peak around their encounter with David. Post that we see he has totally invested in her, perhaps over-invested in her, and hence the stage is set for the final, brutal climax of the narrative when yet again someone he cares about obsessively is in danger and people stand between him and his goals.
The final Coda wraps his tale up nicely - he's not a monster or a psychopath nor is he a hero or even an anti-hero. He's a guy with a particular physicality and state of mind and the true traits of the selfish survivor who makes in the end a simple choice that is completely in character even if many would question his choice set against a world where survival is not guaranteed.
Thematically TLOU puts all its main characters (and by main I mean the leads and notable secondary characters) the question of their existence - "can it all be for nothing?" and each reacts to that accordingly. Some fall by the wayside when it seems it is all for nothing, some sacrifice themselves in the belief it then "won't be for nothing" and in the end Joel, the true selfish survivor, ensures that for him it wasn't "all for nothing" even as to do so he eventually crushes the actions of other characters seeking from their perspective to make sure "it all wasn't for nothing".
TLOU clearly chose a setting and context that allowed for combat, conflict and the struggle for survival and then put in place a consistent set of characters each trying to achieve their own ends, each trying to make sure that from their own perspective they achieve their own goals and that for them "it wasn't for nothing" and their drive, determination to succeed and absolute will to survive determines how successful each is.
There's nothing revolutionary about the characters or writing in TLOU in general terms, it still lags films and novels and short stories for example, but in a medium where most characters are barely more than inconsistent animated cliches with bad dialogue it really does stand head and shoulders over most games efforts at proper, strong characterization.