You can't save your game on to the Amiibo.
Some real usages of the Amiibos are:
1: On Hyrule Warriors you can only get one weapon (the Spinner) by scanning one of the Link Amiibos (or at least 1 of the 2 early ones, not sure do the new lines of Links work).
2: In Twilight Princess HD if you scan the Wolf Amiibo you get in to spesial endurance dungeon (bit like Cave of Ordeals but you play as Wolf Link).
Well that's it.
Besides that you can save some stats in Smash bros on to them, and you can carry some data over from Twilight Princes HD on to the Breath of the Wild with Wolf Amiibo, You get special skins on Yoshi's Wooly World, You can scan up to 5 Amiibos per day on Hyrule Warriors to get random gifts of weapons, materials and money, Zelda themed Amiibos give some advances on Twilight Princess HD (exept Ganondorf that makes game twice as hard)...
Not sure what the new Zelda themed Amiibos do.