Like I said, amiibo just use bog standard NFC tags you can probably buy yourself, they just have their tag set up in a factory and games are programmed to do specific things when said tags are detected.
In the case of this video, what you're doing here is simply downloading an app that can tell your phone to do specific things when your phone's NFC reader detects specific NFC tags. Normally, with a fresh NFC tag, you would set it's identity yourself, but since the amiibo already have their tag IDs set up, you are simply giving them a task to run on your phone, which can be literally anything from immediately changing a setting down to a specific value, to opening an app, to opening a specific webpage.
I forgot to answer this in my previous post, but in regards to "could a new Smash use data from Smash Wii U/3DS?", that depends.
If it's a port of Smash Wii U/3DS, I see no reason why it shouldn't, as there would not be any major changes to how characters or the game mechanics function.
If we're talking a new Smash game built from the ground up, then no, it wouldn't use the character data from Smash Wii U/3DS, because they are completely different games and would be programmed differently. It's likely they would still give the old amiibo functionality in the new Smash game though, even if the old data is not compatible.