Is there an equivalent way to do something like this in Babylon JS:
Babylon has a Solid Particle System and it uses 1 draw call.
What are the main differences between manipulating everything through a mesh and doing it through a shader?
Well kind of. I don;t know how they do it (maybe it is comparable to our GPU particles but instead of emitting a quad they emit a cube)
So like this but have it emit a cube? GPUParticleSystem - Babylon.js Documentation
They have “Several hundred thousand” cubes that orbit a moving target, like a VR controller.
Is there a way to physically interact somehow with GPU particles? Even if its just pushing and pulling them all?
They are based on a buffer so technically yes 
But remember that Unity can use far more ressources than WebGL (threading, directx12, etc…)