Assuming both lenses are being used on the same size sensor, the area in the frame with a 600mm lens should be one quarter the area in the frame with a 300mm lens. The linear dimensions should change by a factor of two, the areal dimensions should change by a factor of the square of two, which is four.

If you are not seeing the same object shot from the same distance appear to be twice as wide using the 600mm lens as with the 300mm lens then something else is going on here.

1. One or both of the two lenses may not be labeled properly. The chances are probably greater that the 600mm lens is not really 600mm than that the 300mm lens is appreciably longer than 300mm.
2. You are using a Nikon full frame (FX) camera and the 300mm lens is a Dx lens. When a Dx lens is attached to an FX camera the camera will automatically crop the image to only use the center of the FX sensor - the part that is completely covered by the smaller light circle of the Dx lens. This will give a field of view with the 300mm lens that is closer to the FoV of a 450mm lens with an image circle large enough to cover the entire sensor. Look at the resolutions of the photos using the two lenses. Are they the same? Or is the image with the 300mm lens much smaller in terms of pixels than the image with the 600mm lens?