I'm loving this story. I've wondered for a while what it would be like to have the legal adoption come first before any relationship-building. I would have preferred Snape to have his canon background, and I'll explain why below.
This is actually a surprisingly satisfying place to end the story, but if you wanted to continue you could have Snape reveal that he knew Lily as a child, before Hogwarts, and that she was his best (only) friend; and Harry believe it's part of the backstory they are concocting. This misconception goes on for a bit as they continue to bond, before Harry finds out that Snape is telling the truth. Harry becomes angry with Snape because he *could* have come to meet him when he was eight and they *could* have had this good relationship all along (and he could have had relief from the Dursleys sooner). The denoument is when Snape confesses to Harry that he wishes he had too.
But hey, I like fluff.
Author's Response: I do plan to continue this, just got a bit stuck on the next chapter. I'm glad you like it.
If you recall, Snape has already said "we say that ... because your mother was an old school friend", which Harry has taken to be part of the story, which would lead up to the scenario you described (which isn't far away from what I had planned, but what I had planned was a lot vaguer, and would you mind if I used your idea?).
I often detract from canon for backstory for several reasons. It keeps me on my toes, and I like to explore alternate ways the characters could have ended up where they do. (I've got some fics with much more intricate backstories in the works.) I like to surprise readers. I feel also that when you write lots of fics revolving around the same characters, you become in danger of writing lots of similar conversations, which also cover the same ground as so many other stories by other authors. I like to make my stories stand out a bit more by adding a twist or two. Plus I have some issues with DH and usually ignore HBP in my fics anyway to allow more flexibility.
Having said all that, I usually include some form of Severus/Lily backstory, because I like the couple and to me it wasn't something from DH, it had been a fan theory I believed in for years beforehand. But unless my stories actually take DH into account (which this one does not), then the backstory of them is not the same as canon.
I hope I've made sense and not confused you! Anyway, thanks for reading!