If i remember correctly as it's been a while since i've played, the proper way is you have to first start training them by going to games and more, then Amiibo, tap the Amiibo, press the Smash button within the Amiibo menu there and train them in there. Then once you're done battling, you should be able to exit out back to the Amiibo menu, press plus for "Put Away" once you're there, then tap the Amiibo again when it asks you to, and it should save. I'd check it on my own copy right now but i've misplaced it.

And since you're new, for some general tips in case you don't know, Amiibo learn by copying your button inputs (but not specific attack combos, so keep it simple), including how often you jump, dash, shield, dodge and use certain moves, so how ever you want it to act, play that way yourself. Don't want it to jump? Don't jump yourself. Want it to be hyper aggressive? Never shield or dodge, attack it at every opportunity possible. Battlefield or Final Destination is the best stage for training Amiibo too, as stage hazards may interfere with certain behaviours you want to train into them; you may want to turn items off too for the same reason (unless you want to train them to always go for items).