The King in the side story called himself "Obayashi", which - taken literally - is indeed "in" K obayashi, as in her name.
This could strongly suggest that White is indeed someone who is/was familiar with the King, and more importantly their fake name - meaning it's likely that they're either his descendant or are Kiyama (the kid).
And since you mentioned it, the chance of Kiyama being Takiya is actually quite big: Aside from the similar appearance and hobbies; the kid appeared to be a Sentai fan, which means he most likely came from the 80s or 90s. This lines up fairly well with Takiya's age (mid 20s) considering that the series takes place in the present day relative to its first chapter's release, meaning that it's currently around 2013-2014 in their world.
We also know that Kiyama isn't the kid's real name, and considering that japanese people (at least in manga) tend to derive fake names/nicknames from their own real names, it could very well be that "Kiyama" is derived from "Takiya Makoto" - Takiya's real name (yes, that is his real name. Yes, I had to look it up).
Putting these two together and speculating that White might be Takiya, we find even more potential (albeit tenuous) connections: The essence of the magic imbuing White's sword is described by Kobayashi as "ominous", and appears to be familiar to her - we can tentatively consider this to mean that the magic is Fafnir's, since Kobayashi is familiar with it and it is often described as giving off an ominous feeling.
We can even go full tinfoil hat and say that "Evil God Dawn" refers to fafnir himself - since in the raws for last chapter, the part in brackets of "Apostle [of the Evil God Dawn]" is written as "jashin don no" - with "jashin" meaning "evil god", "don" being a transcription of the word "dawn", and "no" meaning "of". You could maybe argue that "don no" is actually a subtle reference to the honorific "-dono", which Takiya has used in the past when referring to Fafnir, and that "Evil God" is just an apt description of him which Takiya uses in place of his real name - but I think that's just grasping at straws at this point.