Knighthood has evolved a lot. Believe it or not, there are many European nations that have an unbroken tradition of knighthood, including specific Orders that have existed for centuries. In England, they have more than one form of it, including the Order of the Garter which has only a very specific number of "seats" that it keeps filled. In all of these cases, it's a title that has the much later connotation of what knighthood was, that it was evolving into during the Renaissance.
I agree, most of the general knightings we see in England today aren't what they used to be. To me it almost seems like it's a desperate attempt by their royalty to remain relevant in popular culture. I mean, Bill Gates? Really?
As for the idea of titles and knighthood in the US, there's some wiggle-room here. The constitution prevents the government from officially offering or sanctioning titles. However, there's nothing to stop private citizens from doing what they want in this regard, or private organizations (religious institutions, fraternal organizations, clubs, Orders, etc) from using whatever names and titles they wish. You can still be knighted by a foreign power, but the US government is disallowed from giving you any special treatment due to it. Royal bloodlines can live within our borders and retain their "royalty" in the eyes of the world.
So being knighted by town mayors doesn't carry any force of law behind it, but there's nothing to stop anyone from having a ceremony and using the word "sir" if they want to. In the UK I think you'd still be OK to do this within a private organization, but they have legal titles in their country, so going around calling yourself "sir" in a public context there would be a bad idea.
So I think you can call yourself a "real" knight if you've been knighted by someone else who is a "real" knight, or by one the of the orders or royalty in an existing unbroken tradition or bloodline, or if you're in a private organization that uses the title. The latter is an important point, I think, as that can include the various knightly groups that have sprung up all over the place.
In the end, what is "real" anyway? We are, after all, talking about a word being attached to a human being that is 99.999% genetically identical to every other human being.
