This black cloak is a common form of attire for many characters in Kingdom Hearts. It's role and representation are not exactly understood, but from what characters have worn it, it's assumed that this cloak is worn by those who regularly travel through the Corridors of Darkness.
The standard Organization XIII attire.
Organization XIII
A common misconception about this cloak is that all 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.
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 blindford across his eyes.
The King was another who wore this cloak. He was also seen in a version of "Deep Dive" in his cloak. What's interesting about his cloak is that it is much smaller, and catered to his body. His ears are clearly visible on his head as well.