On the Amiibo Japanese official page, Nintendo has confirmed some limitations about their new NFC figurines, as Amiibo can only store data for one game at a time.
"Amiibo can only store data for one read/write compatible game at a time.
If you wish to use the data from other read/write compatible software, please erase the data in the 'Amiibo settings' section of the Wii U settings menu."
In a nutshell, while Super Smash Bros. for Wii U reads and writes the data on the figures, early titles with support detailed - as Mario Kart 8 and Hyrule Warriors - will only read data from the figures. To clarify, Smash Bros. saves the progress of your Amiibo, while MK8 and Hyrule simply recognises the Amiibo (in terms of whether it's Mario, Link etc) and activate pre-programmed extras, in the form of Mii race suits and items.
The good news is it seems a temporary situation, so things will change once new Amiibo's games will be released. The first Amiibo's wave will be released on November 28th, alongside Super Smash Bros. for Wii U's release date. As we revealed yesterday, 11 new figurines will be available early next year (on January 23th and on February 20th). You can download the new Wii U update (version 5.3.0) that allows you to use Amiibo later this month.