Shifting identities and shifting status are a theme throughout "Emma." Emma herself has a very good notion of who she is, and of her place in the world, so of course she can't imagine being in the same position as Harriet, who doesn't even know who her parents are. As for changes in status, the book opens in the aftermath of Miss Taylor's wedding, which elevates her from governess and companion to new bride of a man with his own property, not a small thing in those times.
As has been pointed out, Emma doesn't have any proof that Harriet is a gentleman's daughter. Emma is trying to elevate Harriet's standing, and in fact, by inviting Harriet so often to Hartfield and singling her out as a friend, Emma no doubt has already given Harriet more status in the eyes of the teachers and other girls at Mrs. Goddard's. And Mr. Knightley is right, there is a real danger that Emma will give Harriet a notion that she is too good for the people she has been used to associate with.
Harriet is not the only young person who has grown up away from home and parents. We learn that Weston's son Frank has been reared by his late wife's relatives and has even taken their name. They are rich people and his station in life is no doubt much higher as a result than it would have been as the son of a military man or tradesman, no matter how respectable.
As a sad aside, we get a glimpse of lifespans and mortality in those days. Both Emma and Frank lost their mothers during childhood. As for Harriet, it is possible that her mother has also died and that her natural father has taken all the responsibility for her upbringing.