I'm not sure what stereotype you're referring to, but the length of Harry's surname reads to me as almost a parody of the inclination to signal egalitarianism. I take it as evidence that his adopted parents (particularly the father) are Very Liberal, but that's all.

I agree with this, and I think it may also have been inspired by/intended as a parody of either:

*The fanfics where Harry finds out he's descended from Merlin or the Founders and picks up a pile of extra surnames

*The use of hyphenated titles for characters like Boy-Who-Lived or He-Who-Must-Not-Be-Named in canon and fanfic

Even if it's not inspired by either of those, the associations made it funnier.

[edited for format issues]