It's Egg Toss Slide. I watched a video of ETS in Brawl because I couldn't quickly find one for Smash 4, and I'm wondering how important of a technique it is for good Yoshi play. It basically just looks like a tool for egg keep away, which could be useful in certain situations. It looks like you would need jump set to a shoulder button to do it, but I'm not really convinced if it's worth it.
Also, to answer the question about the amiibo, I can't say I've extensively trained my Yoshi, but he's been at level 50 for a while and I haven't noticed him use my techniques. I cringed when I watched him fight a level 9 computer...and get 2 stocked. He wouldn't engage in the fight and kept using his double jump over and over again for no reason. He's completely non aggressive for some reason, but he somehow picks up his game a bit when he plays against me. He just has really good defense and perfect shields most things.