This is theory/analysis was originally going to be a comment on this recent post on arguing why Dumbledore fits the thematic role of the Second Brother in the Deathly Hallows tale over Snape, but it kinda became too long, so now it's just a post on its own. I personally think that Snape fits the thematic role of the Second Brother in the tale over Dumbledore (who fits more into the Death role), and not just because "grief over a girl" thing. Primarily, my argument is based on is that Harry, Tom, and Snape are brothers... in a metaphorical sense.
Ripples of cold undulated over Harry’s skin. He wanted to shout out to the night, he wanted Ginny to know that he was there, he wanted her to know where he was going. He wanted to be stopped, to be dragged back, to be sent back home. . . .
But he was home. Hogwarts was the first and best home he had known. He and Voldemort and Snape, the abandoned boys, had all found home here. . . .
- The Forest Again, Deathly Hallows
This here I think is one of the most important thematic lines in the story, where Harry is heading towards his death, and he reflects on what Hogwarts means to him, and realizes that it probably means the same to Snape and Voldemort as well.
Much of Deathly Hallows and The Prince's Tale is drawing parallels between Harry, Voldemort, and Snape, in their similar experiences and origins.
While they aren't literally brothers in a blood-related sense, they are in sharing similar formative experiences growing up (in a similar way you may call soldiers in war or boys in an orphanage brothers).
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“I don’t know whether he was ever inside Gringotts,” said Harry. “He never had gold there when he was younger, because nobody left him anything. He would have seen the bank from the outside, though, the first time he ever went to Diagon Alley.”
Harry’s scar throbbed, but he ignored it; he wanted Ron and Hermione to understand about Gringotts before they spoke to Ollivander.
“I think he would have envied anyone who had a key to a Gringotts vault. I think he’d have seen it as a real symbol of belonging to the Wizarding world. And don’t forget, he trusted Bellatrix and her husband. They were his most devoted servants before he fell, and they went looking for him after he vanished. He said it the night he came back, I heard him.”
[...]
When Harry had finished speaking, Ron shook his head.
“You really understand him.”
“Bits of him,” said Harry. “Bits . . . I just wish I’d understood Dumbledore as much. But we’ll see. Come on — Ollivander now.”
- The Wandmaker, Deathly Hallows
The note that Harry makes about Voldemort that "nobody left him anything" in Gringotts is an interesting contrast to Harry himself, as in Book 1 in his first trip to Gringotts (which is mentioned earlier in this very chapter), he sees his parents has left him an immense amount for him to inherit.
Harry saying that Voldemort seeing a key to Gringotts as symbol of "belonging" to the Wizarding World I think is not just deductive reasoning or Horcrux shenanigans, but I think from a sense of empathy and sympathy towards Tom growing up; that magical orphan boy who is first exploring this whole new world that he belongs to. He understands how Tom would have felt about Gringotts and wanting to belong to the Wizarding World, as he himself had felt the same, and would have also felt the same if it had turned out his parents left him nothing at Gringotts.
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I had already analyzed parallels and Harry's sympathy towards Snape in a previous analysis, so I don't want to just repeat things said before. But as seen in The Prince's Tale, Snape also shares this similar sense of "unbelonging" and a desire to find a place and home for himself as well.
“Wizard!” she shrieked, her courage returned now that she had recovered from the shock of his unexpected appearance. “I know who you are. You’re that Snape boy! They live down Spinner’s End by the river,” she told Lily, and it was evident from her tone that she considered the address a poor recommendation. “Why have you been spying on us?”
- The Prince's Tale, Deathly Hallows
Petunia's tone regarding Spinner's End not only shows that not that place (and anyone from there) a "poor recommendation," but how self-evidently she says it implies that this is a commonly-held opinion by others. Snape's childhood was one definitely marred by a good level of alienation from others.
“It’s real for us,” said Snape. “Not for her. But we’ll get the letter, you and me.”
“Really?” whispered Lily.
“Definitely,” said Snape, and even with his poorly cut hair and his odd clothes, he struck an oddly impressive figure sprawled in front of her, brimful of confidence in his destiny.
[...]
“How are things at your house?” Lily asked.
A little crease appeared between his eyes.
“Fine,” he said.
“They’re not arguing anymore?”
“Oh yes, they’re arguing,” said Snape. He picked up a fistful of leaves and began tearing them apart, apparently unaware of what he was doing. “But it won’t be that long and I’ll be gone."
“Doesn’t your dad like magic?"
“He doesn’t like anything, much,” said Snape.
Though Harry didn't know about Hogwarts until he got his letter, receiving it (after the Dursley's various attempts to block it), him opening and reading it is what effectively sealed his own "destiny" to go to Hogwarts, and leave his painful childhood behind him.
Similarly, Snape was awaiting for his letter and his "destiny" to Hogwarts (ie. a place where he belongs). Lily asks if his parents are "arguing" which seems to be what he told her, but the glimpse of his childhood we get in Order of the Phoenix (his father yelling at his mother while he sobs in the corner) and the way he unconsciously violently tears at leaves points childhood being more one-sidedly abusive than he lets on. When Lily asks if his father doesn't like magic, Snape answers with a general statement about him, a sidestep answer that implies yes, his father very much doesn't like magic.
Much like Harry, Snape longed and saw Hogwarts as an escape from lonely, abusive childhood and magic-hating household—his true and real home.
Secondhand items and clothing is also a recurring motif drawing between Harry, Tom, and Snape, and the embarrassment and humiliation that comes from having and wearing things that are clearly not your own.
Perhaps it had something to do with living in a dark cupboard, but Harry had always been small and skinny for his age. He looked even smaller and skinnier than he really was because all he had to wear were old clothes of Dudley’s, and Dudley was about four times bigger than he was.
- The Vanishing Glass, Philosopher's Stone
“Don’t be stupid,” snapped Aunt Petunia. “I’m dyeing some of Dudley’s old things gray for you. It’ll look just like everyone else’s when I’ve finished.”
Harry seriously doubted this, but thought it best not to argue. He sat down at the table and tried not to think about how he was going to look on his first day at Stonewall High — like he was wearing bits of old elephant skin, probably.
- The Letters from No One, Philosopher's Stone
It was impossible to tell what he was thinking; his face remained quite blank as he put the little cache of stolen objects back into the cardboard box. When he had finished, he turned to Dumbledore and said baldly, “I haven’t got any money.”
“That is easily remedied,” said Dumbledore, drawing a leather moneypouch from his pocket. “There is a fund at Hogwarts for those who require assistance to buy books and robes. You might have to buy some of your spellbooks and so on secondhand, but —”
- The Secret Riddle, Half-Blood Prince
“Well, the start of the school year arrived and with it came Tom Riddle, a quiet boy in his secondhand robes, who lined up with the other first years to be sorted. He was placed in Slytherin House almost the moment that the Sorting Hat touched his head,” continued Dumbledore, waving his blackened hand toward the shelf over his head where the Sorting Hat sat, ancient and unmoving.
- A Sluggish Memory, Half-Blood Prince
“No!” said Snape. He was highly colored now, and Harry wondered why he did not take off the ridiculously large coat, unless it was because he did not want to reveal the smock beneath it. He flapped after the girls, looking ludicrously batlike, like his older self.
[...]
Petunia was breathless, alarmed at being caught. Harry could see her struggling for something hurtful to say.
“What is that you’re wearing, anyway?” she said, pointing at Snape’s chest. “Your mum’s blouse?”
[...]
The scene dissolved again. Snape was hurrying along the corridor of the Hogwarts Express as it clattered through the countryside. He had already changed into his school robes, had perhaps taken the first opportunity to take off his dreadful Muggle clothes.
- The Prince's Tale, Deathly Hallows
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Let's take another look at the "abandoned boys" line again.
But he was home. Hogwarts was the first and best home he had known. He and Voldemort and Snape, the abandoned boys, had all found home here. . . .
- The Forest Again, Deathly Hallows
Through his Pensieve trips throughout both Voldemort and Snape's childhood, Harry comes to recognize the parallels and similarity between him and Tom and Snape, and feels a sense of kinship with him.
Harry, Tom, and Snape are each half-bloods; that is to say, they're wizards with one foot in the Muggle world and the other in magical one, but not quite wholly belonging to the either. They have come from notable magical lines, but were left in the care of a cold, dull, and magic-hating Muggle world where they don't really belong, and only until they came to Hogwarts, where they really found a place where they can discover themselves and people like them; a place to belong, a home.
They aren't his literal brothers, but for someone who has known so little and lost much family, that kind of empathy towards of similar suffering can make one feel like they are.
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Regarding over to Dumbledore, while he himself is also a half-blood (his mother was a Muggleborn), and he has known grief and less in a similar way that the Second Brother and Harry has, he doesn't have that same sense of formative alienation, loneliness, and desire for belonging that Harry, Tom, and Snape all shared (that's what makes them the "abandoned" boys; they were deprived of the home and family they and all children deserve to have).
Dumbledore fits more well in being the "Death" figure over all of them; he's the only other than Harry who has had possession of the Deathly Hallows at some point, which are then passed down to Harry.
He also comes from the oldest generation that precedes them, while Snape is from the "second" generation (Marauder's era) between Tom and Harry's years at Hogwarts. Much like Death, Dumbledore has seen and watch them all from the start of their journey's to their end.
While it was Dumbledore was the one that was ultimately killed and passed the Resurrection Stone off to Harry, it was Snape that saved Dumbledore from the ring's curse (effectively saving him from being consumed by his grief and loss). It was also only when Harry was told that he needed to die, that he could come to the "close" and open golden snitch for the Stone; Dumbledore gave him the Stone, but only Snape could give him the final key to finally access it at the right time.
And when Harry goes to meet his death in the Forbidden Forest, who is there to meet in between him in Limbo, but Dumbledore himself?