I already said why it couldn't:
People are saying that the FastJump is a footstool hop. If you look at the video, you'll see you get the circular ring under Samus that only happens when you do a mid-air jump. Compare that to mario's footstool hop on Samus in the other video. There is no circular ring in a footstool hop. Also, Sonic was doing his downB attack and if samus were to touch sonic one of four things would have happened:
A) Sonic gets footstool hopped and stops his move
B) Samus gets hit by Sonic's move.
C) Both moves collide and you would see the circle form with the "Perfect-Block" sound
D) both moves collide and sonic would have attack cancelled, stopping his move.
Watch the video and see that sonic keeps on doing his forwardB and Samus moves up with the cingular ring under her that is only present with a mid-air jump. Not only that, but a Footstool Hop adds a little glittery effect to the hop for the first couple of frames with no mid-air jump circle.
You make good points, but I still say it's likely a footstool jump. Let's see if I can cover your points one-by-one.
A) Maybe not. We know Footstool Jumps do 0% damage. Its possible they also can not interrupt you in the middle of an attack, perhaps to prevent possible exploits. We don't know for sure.
B) Sonic is charging up the move, he hasn't unleashed it yet. I don't see why it would do damage when it is charging up. You can see he stays in place for a while even after the jump occurs.
C) See B.
D) See C.
Sparkles - I actually see them in both videos on HD. Its harder to make them out on the lower quality .mp4 but they are definitely there. I see no other effect that's on one video but not the other besides the double-jump ring you mentioned.
And now for the double-jump ring. For this I have to dip into my game programming experience and studies of Melee's coding techniques.
Smash Bros uses a control technique to make the controls feel smooth and responsive where, in the first few frames of a move, it can change into another move based on new input or changed conditions. This seems weird but its really vital to make a game feel responsive. In most cases, the player never notices.
A good example of this in Melee is Up Smash attacks. When you do an Up Smash, if you press Up slightly before A, you begin to jump. However, you then press A and it does the spin attack instead, cancelling the jump. This is why jumps don't leave the ground right away in Melee, so you have time to cancel them into something else.
Likewise, if you press A slightly before Up is high enough for a Smash attack, you will begin the animation of a Nuetral A or UTilt, but immediately cancel into an Up Smash instead when it detects that you pushed up the rest of the way in the first few frames.
There are other cases like this all over in Smash Bros, where something cancels into something else if its within a few frames. In fact, Wavedashing is only possible partially because of this control freedom.
What does this have to do with the double-jump ring?
Simple - in this case, Samus DID double-jump. However, within the time frame of a few frames, she was suddenly in a position where should could be doing a footstool jump off of Sonic instead. The game decides maybe this is what you wanted to do, and does the footstool jump. However, the double-jump rings have already spawned, and it may look bad/noticable to have the effect dissapear after a couple frames, so they just let them play out, figuring most people wouldn't notice a detail like that anyway. Chances are that when the double-jump is cancelled you also don't actually lose it (this is a common thing done in cases like this - for example, in a recent game I did I made it so you could cancel a double-jump into a wall-jump if you happened to meet the criteria of a wall-jump within the first few frames of the double-jump, and at that point I give the double-jump back to you - no one ever notices this happens though).
So there you have it. I wouldn't call it proof you are wrong, but I would call it enough to cast doubt on it being a fast jump instead of a footstool jump, especially since we've seen no other cases of a fast jump in any videos with a lot of n00bs jumping around everywhere.