I believe Polka Dot Pirate was made with Gamemaker, which is a 2D engine, so he also had to build up a model/view projection and map his 2D tilemap onto it. I wouldn't be surprised if much of that was done in shader code, I chose to do this all in-engine since I could just use Unity's 3D support. I have no idea how my approach compares, but after some research and fiddling around I settled on using Gerstner Waves and then projecting my flat tilemap around a semi-sphere, moving the tiles to simulate movement of the player.
Hey, just came across this, it's fantastic! I can answer a few questions if you like.
You're right in that it was made with Gamemaker, but I didn't use a shader to wrap the dots around the sphere (I
did use a shader to wrap the clouds around, though). The position of each dot (as well as its alpha) is calculated using trig functions (which is why they stay as perfect circles). Then the waves are done by adding an offset that moves around in a small circle. I'm not sure whether that's the same or different to Gerstner waves, to be honest!