Ironically, I think this Occlumency lesson was one of the most amicable interactions between Snape and Harry. Harry leaves a nasty little injury on Snape’s wrist, but Snape seems unbothered, and doesn’t get angry or annoyed with him. Snape sees a dog chasing Harry up a tree in his memories with his family laughing, and he’s like, “Hey… whose fucking dog was that, man…?” I wonder why else he would comment on it except out of concern. Obviously he doesn’t care whose dog it is because he thought it was cute. Oh, and he compliments him!
OP I've got links to two metas you might be interested in... one in a thread from an ask deathdaydungeon got a few years ago, and one where I couldn't stop thinking about exactly this moment and went off a bit. Both basically connect this moment with the fact that it's at the end of this same book that members of the Order show up at Kings Cross to defend Harry and make sure the Dursleys treat Harry reasonably. Why then and not earlier? Because they didn't know. How would they know at that point, unless another Order member who connected some dots told them to?
tl;dr I think you're right, Snape was concerned. We don't get any glimpses into his thoughts and can only piece things together, but I think he would have noticed more than just the Dursleys having a laugh at Harry's expense while being chased by a dog - we know from PS that before he went to Hogwarts, they dressed Harry in Dudley's old clothes. There's also an allusion to this incident in PoA:
“On her last visit, the year before Harry had started at Hogwarts, Harry had accidentally trodden on the paw of her favourite dog. Ripper had chased Harry out into the garden and up a tree, and Aunt Marge had refused to call him off until past midnight. ”
-Prisoner of Azkaban, Ch. 2
Imagine the nice looking, middle/upper-middle class Dursleys laughing at Harry up in that tree as Dudley's old clothes hung off him. Wouldn't Snape have noticed that too?
Snape is described like this as a kid who's almost old enough to go to Hogwarts:
“His black hair was overlong and his clothes were so mismatched that it looked deliberate: too-short jeans, a shabby, overlarge coat that might have belonged to a grown man, an odd smock-like shirt.”
-Deathly Hallows, Ch. 33
I think he would immediately notice what Harry looks like, when he's roughly the same age:
“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. Harry had a thin face, knobbly knees, black hair and bright-green eyes. He wore round glasses held together with a lot of Sellotape because of all the times Dudley had punched him on the nose. ”
-Philosopher's Stone, Ch. 2
I think Snape noticed more than just that Harry's "family" were laughing at him while he was terrified and chased up a tree by a vicious dog. I think he also noticed and understood the implications of a skinny boy in overlarge clothes and broken glasses that were fixed with tape, instead of replaced. With a young Snape, there's maybe room to try and argue that his mismatched, ill-fitting clothes were a sign of poverty (although it's more likely they were a sign of neglect for a number of reasons - poverty often means shabby clothes but still reasonably fitting ones, Snape's clothes were repurposed adult's clothes not used children's ones which is also unusual, and we see Harry's magic reject an ugly sweater Petunia tries to put on him so that it shrinks and is unwearable so perhaps a young Snape neglected himself as well because it was what he learned and accepted the clothes he was given). With the young Harry in the memory Snape sees, the Dursleys are standing right there for comparison, so Harry stands out in his overlarge clothes and broken glasses against the very people he lives with who have well-fitting, nice clothes. If we know anything about Uncle Vernon, the guy who liked to go outside and talk loudly to his family about his new car so the neighbors would hear, it's that he liked to show his status and cared what people thought about him, as did Petunia. Harry's neglect would have been glaring in that memory, not just on its own, but seen side by side with the people who clothed him in obvious hand-me-downs and didn't bother to fix his glasses while they looked well-kempt and cared for themselves. The glasses are a loaded thing - broken multiple times, because Dudley punched Harry, someone like Snape who understand neglect and abuse firsthand would like wonder if they failed to be replaced out of neglect or because there was no point if they'd just get broken again.
Snape must have noticed, as he's an observant, intelligent character. In this moment in OoTP he's suddenly more patient with Harry than we've ever seen him, and doesn't respond to Harry's reflexive curse as if a mini James was attacking him, but instead is uncharacteristically unbothered. I think it's the first time he sees Harry as Harry, and not an extension of his father. And I think that when Harry breaks his trust by entering the pensieve and seeing SWM, Snape realizes how dangerous it is to himself to have any kind of personal connection with Harry.
This is great analysis and this part always makes me so sad because it makes me imagine an alternate universe where Snape was able to overcome his antipathy towards Harry, see him as his own person, and they actually bonded. Snape could’ve been an adult in Harry’s life. It would’ve been so nice for both of them.
