A fife usually sounds breathier to me than the wind instrument in the song, so that I definitely would say the instrument used was a soprano recorder.
Here's what Wikipedia said about the music composed for the Final Fantasy Crystal Chronocles series:
"The soundtrack to Crystal Chronicles was primarily composed by Kumi Tanioka, while music programmer and arranger Hidenori Iwasaki provided one additional piece of music. Tanioka has described the musical style for the soundtrack as being based on 'ancient instruments'. The soundtrack makes extensive use of many medieval and Renaissance musical instruments such as the recorder, the crumhorn and the lute, creating a distinctively rustic feel."
Final Fantasy Crystal Chronicles - Wikipedia, the free encyclopedia
You can compare the video I posted above with the following, which briefly shows someone playing a renaissance fife at the very end, demonstrating what I mean about the breathy sound of fifes:
Renaissance Fife - breathy!