Going into the NX, some of the third parties that are close to Nintendo will definitely use existing amiibo and some will create 3rd party amiibo. I don't think there will be stronger integration then what's been done.
If a multiplat game comes much later to the NX, then it will probably use an amiibo. If not, it'll probably cause an immediate uproar from other platform owners and some NX owners. The only exception to the that would probably like if a game like FFVII R launched at the same time with the PS4 and NX and it used pre-existing amiibo (Mario, Link, Ike, etc.) for aesthetics like costumes
* Saw this earlier but had things to do :p