There are several Leviathans mentioned in lore.
First, there’s the ship. (Not actually first but well start here.). In one of the pair cards it shows a large creature, presumably a leviathan, on a cabal planet with Cabal looking at it either eating a city or falling from the sky and sliding across the planet. It could have then been weaponized to the ship we have now.
Secondly, there’s the Nine. Descendants of Leviathans from planets like Titan, literally mentioned. Xur is their agent and the actual Nine could be actual Leviathan minds projecting this emissary to us. Still a lot to find out.
Lastly, the most interesting, is the Leviathan that Oryx and his sisters encountered while fleeing the Helium Drinkers. The Leviathan, described as a student to the Traveller, warns them not to delve deeper into The Deep and warns them about the Worm Gods. After gaining power from the Worms, the three sisters attack this Leviathan that was deep in the seas of their home world. I don’t believe it actually referenced an ocean but it was protecting The Deep and was attacked. It could have died or could have been killed in space while Oryx and his sisters were aboard “The Needle (?)”. Could have crashed into the Cabal planet.
The Leviathans are like the old Illuminati, they want what’s best and fight for the light. So they reward us to help fight the darkness. His explains Xur and now explains why this Emissary gifts us weapons for our abilities.
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Found this to be interesting One idea for the Leviathan's origin, according to artist Dorje Bellbrook, was that the ship was inspired by creatures native to Calus' homeworld resembling large "land whales" that would travel along the ground and swallow anything caught in their path.[5] It is unknown, however, whether this is canon information or not.
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The thing is, it’s a game. So things are named a certain way. It could be just a name for these creatures which could have been powerful worms that grew to enormous sizes. Inspired by these large creatures that would eat entire cities, he could have named his ship after them. But he was exiled onto the Leviathan, not onto a ship he named.
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A younger Leviathan protecting something. It could be a doorway for Savathun to enter back into our solar system. It could be more Worms that the Leviathan doesn’t want to be met by the other Gods or the two remaining Hive Gods.