Black coat

The Black Coat (sometimes called an Organization coat, or incorrectly as a 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 available to those who wield darkness.

It is assumed that this coat is worn by those who regularly travel through the Corridors of Darkness. It appears that the black coat is granted to those who travel through the Corridors of Darkness as a symbol of their ability to resist the potential corruption in doing so.

Somehow, when Riku and King 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 coat disappears, such as Riku's collar, jeans, and sneakers in his second attire, or Mickey's trademark white gloves. All these hint that donning the coat transforms whatever clothing the coat's bearer is wearing under it into the black gloves, pants, and boots seen on all the coat's wearers.

Organization XIII
A common misconception about this coat (besides its being frequently called a cloak) is that all its wearers are members of Organization XIII. It is true that all Organization XIII members wear these coats, though. The form differs for each member; each member wears the main zipper at a different height, and some members' coats are thinner in certain areas. For instance, Axel wears his coat very tight against his arms while Xigbar's zipper is unzipped at a higher level than the other members' coats. Additionally, Vexen's coat extends all the way that it covers his feet like a gown, compared to the rest. Demyx has a stylized shoulder gauntlet formation on his coat, making his appear more like a rocker's outfit. Roxas wears what appear to be black sneakers, while Larxene and Xion wear heeled boots.

Other wearers
The first person to wear this coat outside of Organization XIII was Riku. He is first seen in this coat fighting with Roxas in the "Deep Dive" movie. Along with this coat, Riku wore a blindfold across his eyes. Riku was given his Black Coat by DiZ so that he would not be tracked down by Organization XIII.

The second person to wear the coat was King Mickey. He was seen in a version of "Deep Dive" wearing it. He was even seen in Sora's Awakening scene wearing the same coat, while he was coming to Sora and company's aid. What's interesting about his coat 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. When Goofy is injured after the fight with Demyx, Mickey removes and discards his coat and charges into the Great Maw.

In Kingdom Hearts coded, a mysterious figure is seen in the black coat. In the fourth episode, it is revealed that the man is actually Jiminy's Journal taking on the form of Riku in his coat while helping Sora. The Journal mentions having met the Data Sora in Traverse Town first, leading to speculation that another as-of-yet unknown person in the coat has appeared.

In Kingdom Hearts 358/2 Days, Moogles also wear the black coat, as a sign of their services for Organization XIII.

In Kingdom Hearts Birth by Sleep a man in a black coat is seen near Ventus (who is wrapped in a white cloth) on Destiny Islands.

Trivia

 * The black coats were originally designed for Roxas and Riku in the secret video Another Side, Another Story... simply as a way to disguise them while they were still being designed for Kingdom Hearts II. It was then given to Xemnas in his optional boss battle in Kingdom Hearts Final Mix and in Deep Dive, starting the trend of all Organization members wearing it. It was again used simply to hide his face here.
 * In Kingdom Hearts Birth by Sleep, Master Xehanort is seen waering the coat.