Personally, what I love about BlazBlue is how readily it straddles the line between "serious" and "parody". It's a similar praise I have for the Metal Gear Solid franchise, in that the writing keeps me hooked not because of how good it is (to be honest, a lot of the writing in both BlazBlue and Metal Gear Solid is hilariously "bad", or at least poorly edited) but because of how, as you put it in your OP, interesting it is. BlazBlue is like a simultaneous love letter to and satire of anime, but it manages to pull off both positions really well, simply because of its self-awareness.
I'm a strong proponent of self-awareness being a huge factor in whether something is "bad" or qualifies as "so bad it's good"; this is a tangent, but to me the writing of a game like Final Fantasy XIII becomes tedious because it exhibits no self-awareness for how ridiculous its story or characters are. Everything is presented in a straight-faced, matter-of-fact manner, and while that's not a bad thing in itself, it doesn't help the game's position when everything it's presenting to the player is nebulous at best. BlazBlue manages to side-step that entire issue simply by knowing it's a silly game. Clearly, that won't make it tolerable for everyone. It's still a niche product, and has relatively limited appeal. But by playing to its strengths, it manages to make the most of the absurd and occasionally nonsensical things it presents to the player, and it allows for things like the branching narratives resulting in the "true ending", "bad ending", or "silly ending"--none of which feel out of place within the context of the game itself.
As far as how it impacts or is impacted by being a fighting game as well, I think it honestly helps the narrative that they're trying to tell. In many games, pacing is a legitimate issue not solely because of the writing, but also because of the encounters and how the developers try to account for the player's potential skill. Because many games present the protagonist as being a one-man army who is capable, and indeed required to mow down hundreds of enemies thrown in their path. Fighting games are one of the few genres that have the unique ability to side-step that without any strange contrivances or compromises, by the nature of them being 1v1 encounters (or upwards of 3v3, depending). BlazBlue, for all of the war/rebellion/conflict contained within the narrative, is more about the characters than any big event setpieces, and being presented as a visual novel and fighting game allows it the freedom to explore every nuance of those characters.
Now, admittedly, it's all still highly ridiculous. BlazBlue intermingles small comedic sketches almost directly beside "serious" drama, often with little or no interlude. And as much as it seems to be taking satirical swipes at anime, it's no stranger to classic anime tropes such as fanservice of characters either. It could easily be accused of having tonal inconsistencies, and those accusations would hold a lot of weight. Subjectivity makes the final call then; for me, it creates something I want to see out to its conclusion because it's a unique change of pace both for me and for, I assume, fighting games. But by apparently putting its central focus on the narrative, or at least a heavy focus, it gives me something to feel invested in. I mean, it's the same reason I read any visual novel, or play any RPG. They put the effort into crafting a narrative, and while they way they did it appeals to a very specific group of people, I always feel more interested when it feels like the developers themselves were interested.
Sorry for any rambly bits, it's 1:30 AM as I'm typing this and I ramble even when I'm not tired.
As for Injustice, it's definitely a different context but another where I would say the writing and story isn't so much good as it is interesting. I like the premise of it to begin with, to be honest, because it shakes up an established bit of canon in a pretty dramatic way. I won't dwell so much on the actual writing of this one, though, because it's... well, I'm just not sure how interesting it would be to people who weren't already interested in the DC Comics characters to begin with. I assume the same position I take with most video games, in that the writing itself isn't particularly good or bad, but merely passable. It conveys the story they want to tell in an enjoyable manner. Which probably sounds fairly more boring than I mean it to, so I'll sum it up as "it did an entertainment".
What I like about Injustice is almost entirely granted to its presentation. I'm not one to gush over how "cinematic" a game is, but I was genuinely impressed by the way the cutscenes were used to smoothly transition from one fight to the next, even between stages. It's not something that I think I've ever seen in a fighting game, though I can't say it hasn't been done before, and I feel like that's a decent way for a more "traditional" fighting game than BlazBlue to tackle having a focus on an actual single-player narrative. If I could remember them more concretely, I would make comparisons to the Dragon Ball Z: Budokai franchise and how it presented its single-player narratives here. I remember they switched things up pretty extensively from game to game, but I don't remember exactly which game did what and all of them were a bit slowed down by mucking around in between fights.
Being so cutscene-reliant does subject Injustice to the same issues that most other modern games tend to have, especially with regards to shot direction and character modeling, scene setup and all of that other good stuff, but it gives it the same strengths those games have as well. It allows for a more seamless story experience, and it allowed the stage/world designers to really flex their creativity as they came up with entire transition scenes and areas, things which remove the stark segmented and disjointed feeling that I always found present in older fighting games.
Obviously, those are by no means the only ways for fighting games stories to be tackled. There are many methods of story-telling alive and well within the games industry these days; at this time of night I can't even fathom how to transplant them onto the fighting game formula, though I can come up with plot/narrative ideas. But I think a simple determination for how fighting game stories can be better just depends on the writers involved. I think a story can only be as good as the writer is invested in telling it. With fighting games, especially before the last few years, I just don't think many developers thought people cared about having stories in their fighting games. When those impressions can be cast aside, we can see a further influx of fighting games which have narratives crafted for people who enjoy the single-player experience.
