The Kraken is a legendary giant squid said to be able to sink entire vessels with ease. He does not appear in either Endless Ocean game; instead, a similar creature of his namesake, Kraken Jr., appears in Endless Ocean: Blue World.
The Kraken is a mythological giant beast that is said to dwell the oceans and attack ships that encroach upon its territory. It is often depicted as a massive creature bearing morphological similarities to a squid or an octopus, and researchers believe that reports of this creature were based upon early sailors' sightings of the giant squid.
Early descriptions of the Kraken gave it a closer resemblance to a creature with traits of both crustaceans and large whales, but it was soon given the famous appearance of a giant cephalopod attributed to it today.
The Kraken originated in Icelandic mythology and was not given any description beyond its fearsome jaws, massive size, legendary aggressiveness, and its tendency to lay at the bottom of the ocean. However, researchers have noted that signs of the Kraken's appearance are very similar to signs of undersea volcanic activity around iceland (sudden bubbles in the water's surface, the appearance of new islands, etc).
Although the Kraken does not appear in either Endless Ocean game, he is mentioned in the "Sperm Whale vs. Giant Squid" scene which you watch with Hayako in the Zahhab Region Depths (this is the same scene you encounter during the story's main plot). He is also mentioned in the trivia for Kraken Jr.