The easiest thing would be to do it with real lighting (spot light). Unless you really want a retro look.
I think those old games used stencil buffers, which aren't supported in Godot and aren't used as much in modern games.
The other option would be to do the fog with a custom shader. The lighting would still be real-time, but you could mask out an area (the volume of the flashlight) and use this to remove the fog. I think this would be the closest to that look.
But it's a little difficult to define masked areas. I guess the fog could be a full screen post process effect (by reading the depth buffer) but you'd need a way to set the area of the flashlight. My guess is that in that game the flashlight is not really 3D but just a 2D shape. This would be easier to send to the shader, as it can just be another texture. But I've never tried this, I'm just sort of guessing. I would suggest you test it using real lighting first, as you may be able to adjust the spot light properties enough that you can get the look you want without complicated shader coding.