Is that both of them (blue and green)?
I suppose you mean the large beam and the side ones? I didn't watch the whole video.
I can't see the individual sprites in the blue beam.
The big beam could be just an animated, moving axis-aligned rectangle. Technically, there are multiple approaches that would look identical, for example one big texture mapped rectangle with animated texture coordinates or several perfectly tiled and moving replicas (maybe a triangle strip) of a large sprite containing one "wave".
And what is the logic behind the sprites in the green beam? Does each sprite follow the one preceding it? And how is it that both beams keep their stream-like fashion no matter how the player moves? Thanks again.
The small "beams" contain very fast large sprites which appear to be aimed at a point about one screen height ahead of the player's ship, fixed speed and not guided; they form streams because they are emitted so fast that the player cannot move far enough for consecutive ones to lose overlap. Compare to the bursts of fast and straight projectiles shot by some of the bosses in the video: with larger sprites they would look like beams too.