I didn't recognise it, although it seemed quite familiar and vaguely Final Fantasy'ish.
However, I just used my Shazam app and it appears to be 'Finders Creepers' by Paddy Kingsland on the album Kpm 1000 Series: Storytellers (1984). So not Final Fantasy at all but a style that isn't too far off some of the compositions we love.
I have no idea what this is. Paddy Kingsland did a lot of music for radio, television, or commercials, so it might have been used in one of those mediums. I'm not sure where you encountered this track, but if the video claimed it was from Final Fantasy then they might be mistaken.