This isn't 8-bit. This is pixel art. The title is misleading, but hardly anyone differentiates between the two nowadays.
The story is all right. Mario could have been smarter and used Yoshi to farm eggs, thouh changing the story for hindsight would ruin the premise. There was also a typo I noticed, as well as the incorrect usage of binomial nomenclature (two-word Latin classification) which doesn't even need "avis" since that denotes class and not speciation.
The animation is what makes this shine, though I would nitpick about small things. Why not have the yellow bird actually speed up? It seems like they all went the same speed, and the blue bird would probably been interesting to see be duplicated to blanket the sky. Again, those are nitpicking suggestions rather than complaints. The effects used really shows how fluid everything goes.
I can't complain about the music, though none of the audio was 8-bit.
I don't normally give the its own category, but pacing is what seems to be the problem here. Not much is going on.
For the characters themselves, you get three different Mario characters with their own personalities, a fourth one if you consider the sun. The most you get from the Angry Birds is the blue bird being attentive and calling out to the red bird. Character differentiation isn't easy in a project like this, but even abilities show where they differ. As for characters that would have been nice to see in the Mario side, I can think of mostly enemies. Black bird and Bob-Omb, Boomerang bird and Boomerang Bros., and maybe even Giant Mario and Big Brother Bird would be nice match-ups, even if that sacrifices a true 8-bit flash. Sprite animation is sprite animation.
Overall, this is a good piece of work, and it's enjoyable. Don't take my critiques the wrong way.