If you go to the end of the movie, you see that when Aladdin frees Genie, his bands break off, but at the end of the movie when genie comes back to see them the bands are back on.
I noticed that, too. I think it's because the genie looks so much better with the cuffs there. If you notice, during the entire second movie, he's got the cuffs on. Having the whole scene where the cuffs come off was just stupid, if they were going to have the genie have them on later to be aesthetically pleasing.
This is so easy to see. It has to be one of the best known slip-ups ever. Everyone notices this one. Another thing you can pick out is in the next two Aladdin movies (Return of Jafar and King of Thieves) the Genie is still wearing the gold bands around his wrists.
I believe the bands coming off was merely symbolic. The genie is able to alter his appearance whenever he wants for dramatic purpose or to be funny. He made his bands disappear himself. Later they are back just because that is how he normally looks.
The bands disappearing when Genie is released are symbolic - they represent an eternity of solitude and serving and now he has been set free of it the bands go too. The reappear in Genie's arm again for the next two movies because everyone knows that genies have wrist/arm bands and they would look stupid without them!
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