Oh, and by "first" video you mean the video posted first in this thread, not the video labeled #1... because in that one, Sonic's two KOs were jumping into DK's Fsmash >.>
So in video #2 (ie. on FD), it's quite interesting. First stock: DK throws him off the stage, and Sonic's in a position with a ton of options. He was high enough to spinshot or SD > jump-cancel either at the edge, or to a walljump. He could have used HA in position to either safely bounce off the FD wall/edge, or safely recover with spring
or jump>spring if the HA doesn't lock on. He could have safely used jump > spring under the walljump-area, which DK wasn't in a position to attack easily. Anyway, I could rag on all day about the options he had, of course hindsight is great (as well as being removed from the intensity of a match lol), but I cringed at first viewing, because Anther chose the most dangerous option, out of a large variety of choices. He drifted in his tumble state until he was dangerously low, then used his 2nd jump (I think), and sprung towards the edge. The DK might've been able to simple edgehog from there. If Anther'd at least given himself the room to walljump as an option, I'd have less of a problem
Now, the
really interesting situation is his third stock. Sonic double-jumped > ASC onto the stage, it became an SDR and he rolled around. However, he slid off the stage during the "skid stop" from SDR. The skid-stop part is still considered part of the spindash, so it didn't restore his 2nd jump. If Sonic had just finished the SDR on the stage, then walked off the edge, he would have been able to use his 2nd jump. I think this actually caught Anther by surprise, which is why he didn't spring until late. Just a guess though.
Neat little note: using SDR's turnaround to avoid that dire skid-stop state is fantastic. Furthermore, ending a fast SDR with a turnaround at the edge of the stage can actually let you fall off the edge facing inwards, into an immediate edgegrab. It's pretty neat ^_^ But I digress from a more important point.
Sonic mains beware: Let's say Anther
did end his SDR on the stage. That is, he double-jumped > ASC > lands in an SDR. He could walk around the stage all he likes, but if he were to use side-B or down-B,
it would still lack the 2nd jump. You have to actually jump, get hit, or even just walk off the stage, for our spindash's "jump data" to get restored properly.

just something to keep in mind. It's pretty upsetting to walk around, down-B off the stage edge, and then discover you can't jump-cancel it.