Godzilla 2’s novelization gave some insight into Kong’s relationship with the rest of the verse.
Basically, Kong is a kaiju like the rest. Of course, and he’s been hearing the Alpha’s call his entire life. But he ignored it because he’s more or less content to stay on Skull Island on top of being really pissed off at the skullcrawlers. But after the events of the movie he’s ready to leave the island due to a combination of A) a new alpha being established (It was kinda up in the air for a few millenia between Godzilla, Ghidorah and MUTO Prime), B) the various kaiju beginning to migrate to Skull Island and Kong not liking visitors (He’s fine with humans as long as they don’t start dropping bombs), and C) the Alpha’s Call is awakening the skullcrawler horde at an accelerated rate, and Kong really doesn’t like that.