I don't know about knights (how closely the mythological knight conforms to real knights, or how real knights conformed to equestrians), but here's what we're talking about with regards to Roman equites. The Latin for a member of the equestrian class, which is often translated "knight," is "eques" is a horse-dude. The first definition the OLD gives is just "A horseman" without military or political connotations. The political connection is only the third definition.

So, because Romans organized their political society in part on military lines, the centuriate assembly, the assembly that elected magistrates with imperium (or in other words officers of high military command) was organized along imagined contributions individual members made to the military (and Roman legend explicitly justified the inequity of the voting classes by reference to military contribution). Equites were people with sufficient wealth that they could be expected to supply and support not only themselves but also a horse for military service. By the late Republic, if these voting classes ever reflected real burdens of military service, they had stopped. By the time of Marius (c. 100 BCE) the Roman army had become professionalized, with the state paying to support and equip soldiers drawn from poorer classes, instead of relying on citizens contributing willingly out of a sense of civic duty (there are indications of this move coming for awhile, though). By the time Cicero was going on (and on and on) about the concord of the orders, the equestrians became largely identified with the economic class of Rome, making money in the provinces and such. Categories of course break down on close examination of individuals - the senate drew its members from the equestrian class, and a military career was necessary for most public office (and so membership in the senate), so there would be a number of equites in the military at a given time, but then again a lot of these would have had political posts - Cicero for instance worked on his commander's staff during his first campaign.