It really looks like there's only 3 'types' of effects in there.

1) Weapon sweeps.They're just 2d textures, a couple look like they're animated with a couple frame growth, fixed shape sweep and couple frames of vanish. It's either created directly as another layer in the character animation, just drawn in a different draw pass. Or, more likely they're pre-canned textures on a quad or fixed shape ribbon that get their leading edge attached to some animated bone on the animation so that it lines up with the sword. This type of thing also goes for the wings in the intro at 5sec, as they'd just be another animated texture like the main character, but drawn in a different drawing pass to give them the glow. It also goes for some of the bigger effects in the super attacks, where the sweeps/hits follow the attacks animation perfectly.

2) Fixed, ranged attacks. Here, i'm mostly talking about the two or three attacks from walkur around the 43sec mark. These are effects with no randomness, and what appear to be some interesting weaving/splitting/curving going on. It looks like particle emitters that just drop a particle each frame that are attached to a projectile. The projectiles have very specific motion patterns, that could be calculated on some formula/bazier curve but considering it's a fighting game, i'd kinda expect their motion to actually be coming from a keyframed animation just so that it's frame-by-frame accurate where the projectile will be.

3) Just about everything else seems like a standard particle effect with a single texture. Some of the hit flashes look like they're playing animations, others are just throwing particles off with random emitter settings in spheres and cones.