This, I think, is the ground of the strange 'relatability' of these globally popular novels: not class, or race, or gender, or school experience or anything like that; and neither because of any quasi-Dickensian textual campaigning against social injustice, creditable though that aspect of the novel-series is. It's that Rowling says to her child readers, repeatedly and eloquently: you are kings in disguise. You possess magical validity and force. And her child-readers grok it, because kids understand the Scottian insight better than adults do. Maybe that's because they are closer to the time when all human beings share perfect, imperial elevation and power, when the whole of creation bends its efforts to placating and maintaining them -- when we are babies, of course. Or maybe it is a more Chestertonian 'old religious conception', the same numinous if unconscious awareness that Wordsworth ascribes to childhood in the Immortality Ode. At any rate, it goes some way to explaining (I think) why Harry has to be the central character, rather than Hermione. Hermione is too obviously special: too clever, too multi-talented and self-disciplined and grounded and so on. Potter is the chosen one not despite but because he is so ordinary; because (the novels are saying) mere common ordinariness, like yours, like mine, is the absolute ground of magical royalty. We are all kings in disguise.
And the stamps at the top of this post? They're there because I like them, and because they insinuate an actual monarch into their top right hand corner. But they do bring out one related point: the equally popular, equally enduring Narnia books say the same things, for (where Lewis was concerned) equally Chestertonian reasons. Lewis's ordinary English children are kings and queens of Narnia, not because Lewis thought representative parliamentary democracy delinquent and wicked, but because his faith told him that we are all of us, the entire demos, kings and queens of Narnia.
The most remarkable thing about the Harry Potter novels is that almost everyone gets a moment of supreme dignity: Harry, Hermione and Ron, certainly; Neville being the one to wield the sword of Gryffindor; Snape's memories and his death; all of the Weasleys, many times; Fleur Delacour's commitment to her husband; Hagrid's care for his animals. The list goes on. I would submit only that this dignity becomes possible and active because irrevocable death is on the table, and a possible fate for everyone. Because it is a closed world not interested in future sequels (given the environment of narrative fiction at the moment, we should be supremely glad of this), characters' fates can be varied.