Godzilla vs Kong sets up a new story for Kong in the MonsterVerse, though it's unlikely Kong 2 would be Kong: Skull Island 2. Although Adam Wingard's movie leads with Godzilla in the title and is fundamentally about both monsters in Warner Bros. and Legendary Entertainment's franchise, the film is really Kong's story; from the opening shot of him on Skull Island to the ending scene of him in the Hollow Earth, Godzilla vs Kong is about Kong finding his place in the world - and, ultimately, his home.
Kong's story in the MonsterVerse following the events of Kong: Skull Island, which took place in 1973, is largely unknown. Monarch eventually returned to the island and set up a research station, which later became a biodome facility used to contain Kong. Over time, Kong grew bigger and the biodome could no longer keep him at bay. Coupled with Dr. Nathan Lind and Apex Cybernetics' plan to explore the Hollow Earth, Kong was taken from Skull Island and brought to the Hollow Earth entry station in Antarctica.
From there, Kong entered the Hollow Earth and discovered his ancestral home - and throne. After battling Godzilla one last time, as well as teaming up with the iconic monster to defeat Mechagodzilla, Kong is now free of being Godzilla's target; neither monster is a threat to the other. While Godzilla remains on the surface, patrolling his domain in the water, Kong is now King of the Hollow Earth. So if there were to be a Kong 2 film, it wouldn't be Kong: Skull Island; it would instead follow the primate in a Kong: Hollow Earth-type of story.
Up until Godzilla vs Kong's release, Kong was never a king, despite what Hank Marlow said about him ruling over Skull Island in the 2017 movie. However, that has now changed, seeing as he is officially King Kong of the Hollow Earth. His ancestors seemingly defeated the other monsters of the Hollow Earth and used the axe - the one made up of Godzilla's dorsal spikes - as their weapon, just like Kong routinely used a spear on Skull Island. Given the throne he sits upon and the fact that Monarch has set up a new observation station in the Hollow Earth, Kong's sequel movie would presumably follow him in this new region.
In a way, Kong 2 could be somewhat like Godzilla: King of the Monsters, except with Kong battling for dominance in his new domain. There's still so much of the Hollow Earth that remains a mystery, including which monsters outside of the Nozuki and Warbat species still live down there. Kong 2 could easily dive deep into exploring the subterranean world, Kong's time as king, and what other monsters live down there and which of them pose as a challenge to Kong.