[mugimugi]

It doesn't make sense for Ougi being the Darkness. There's no point for it playing human and interfering with everything like Ougi does. Especially Ougi saying that what she's doing is similar to what the Darkness does is the biggest indicator that they are not the same.
I don't think Ougi's "job" is something she has to do but rather wants to do and her use of this phrase of "punishing liars similar to what the Darkness does" was situational.

You shouldn't read the following spoilers @Zugspitze since you seem to be anime only viewer.

Spoiler for Novel contents mentioned here:

It seems more like that Gaen seems to have some sort of goal and either the "opponent" tries to counteract which seems to be out of Gaen knowledge range or they know about each other and try to smart each other out. At first I though that Ougi might just be Meme's familiar since he had shown the tendency to name something "Oshino" (Oshino Oshino aka Shinobu..) and Shinobu mentions that familiars are to bound to the master by their name, but now I am much more "convinced" that this is more about alternate timeline/ time traveling people interfering or changing/balancing the current timeline.
This could be the source of Gaen's and Ougi's knowledge and also be a reason why Gaen's plan are able to fail when Ougi interferes, since Ougi isn't part of current timeline. Another indicator is that all the onmyouji disappear, apparently by assault with Ougi being the culprit. The onmyouji are like Gaen's personal league and one after another is taken out apparently. This ain't about life and death of the current living cast though, since everyone seems fine in Hanamonogatari. The "life and death" situation mentioned there could relate to Koyomi getting "killed" and trying to save him of he needs to be saved or that grand scheme which might affect people's lives or the timeline in some way.
Maybe it's really about trying to change or balance something and the other party tries to make that not happen.
The Koyomi being Ougi theory makes sense, considering that he's know being played by Gaen as well, and at this point even lethally taken out. He is also turning into a full vampire which would allow him to change his appearance (and might explain why Gaen tries to counteract that with the sword, everything leading up to that seems to have have the purpose of Shinobu not wrecking the world for that act/cutting the link to Shinobu. The latter would also prevent him from buffing his vampirism even more which leads to his full vampirism), and if he's from the future/alternate timeline then he would know about everything and possible be the party that tries to undo/interfere with Gaen or whatever else is going.