As the others said, it's a matter of reference according to whether you're considering the target or the destination:
Amène-moi ta sœur, je dois lui parler. (Bring your sister here, I need to talk to her: from there -> to here)
Emmène ta sœur au jardin zoologique. (Take your sister to the zoo: from here -> to there)
If that's easier for you to grasp, the same distinction exists with émigrer (emmigrate) and immigrer (immigrate): both deal the concept of changing country (migrer), but émigrer is leaving one country, and immigrer is entering the other. While you do both at the same time, your country of origin will say you emmigrate while your country of destination will say you immigrate.
Again, since it's a matter of reference, the difference is subtle and can show which part you want to emphasize:
L'homme de mes rêves m'amène sur une île
Puts the focus on the destination, so the island is what's important, while:
L'homme de mes rêves m'emmène sur une île
Puts the focus on sweeping you away from where you are now, and the fact that it's to an island doesn't really matter. Considering the island is explicitely specified, and the source is not, I would tend to think the first makes more sense; but that's not to say the second is incorrect.
As for the original song, quite honestly I can't make it out for sure whether Alisson Adams Tucker says one or the other. Sure, the subtitle lyrics say emmène but that's just what whoever put the subtitles thought.