Black coat
This black cloak is a common form of attire for many characters in Kingdom Hearts. It protects the wearer from the darkness in various ways, and in some cases, from detection methods availible to the those who wield darkness. It is assumed that this cloak is worn by those who regularly travel through the Corridors of Darkness.
Somehow, when Riku and Mickey cast their cloaks off, the black gloves, pants, and boots worn under the cloak disappear. Furthermore, any trace of the wearer's clothing under the cloak disappears, such as the Riku's collar, jeans, and sneakers in his second attire, or Mickey's trademark white gloves. All these hint that donning the cloak transforms whatever clothing the cloak's bearer is wearing under it into the black gloves, pants, and boots seen on all the cloak's wearers.
Organization XIII
A common misconception about this cloak is that all its wearers are members of Organization XIII. It is true that all Organization XIII members wear these cloaks, though. The form differs for each member; each member wears the main zipper at a different height, and some members' cloaks are thinner in certain areas. For instance, Axel wears his cloak very tight against his arms while Xigbar's zipper is unzipped very high compared to the other members. Additionally, Vexen's cloak extends all the way that it covers his feet like a gown, compared to the rest.
Other wearers
The first person to wear this cloak outside of Organization XIII was Riku. He is first seen in this cloak fighting with Roxas in the "Deep Dive" movie. Along with this cloak, Riku wore a blindfold across his eyes.
The King was another person who wore this cloak. He was also seen in a version of "Deep Dive" in it. What's interesting about his cloak is that it is much smaller since it's catered to his body. Even the hood is sewn just for Mickey, with places for his ears.
