Welcome to the forums @Zorochase!
I think how Petscop (the video linked) creates that effect is that it uses a Spotlight above the player and has all ambient light disabled rather than using fog. The reason I think this is that the light falloff is uniform from the player radius, while if it was fog then it would be stronger on the back than in the front. At least, I think that is what is happening and is how the achieve the effect.
I think the fog is always relative to the current rending camera in Godot. In Godot 4.0+ there will be ways to control volumetric fog using shaders, so once its released it should be possible to tell the fog to gather around a certain point like a character, but I do not think manipulating the fog like that is possible in Godot 3 (I could be wrong though!)
