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We can't know, but chances are that you are not properly zeroing out your shape group positions and scale or transforming the layer.
If you create an ellipse the likelihood that you will create the path precisely in the center of the composition frame is very low. You can check this by pressing the U key twice to reveal the modified properties. It's highly unlikely that the Ellipse 1 transform position value will be 0,0. Let me explain why this is important.
Vector paths are always based on the layer size. If you create a solid, move it's Transform>Position to one side and change the Transform>Scale to 150, 25, then create a mask path and copy it the vector information that is copied is not modified by the position, rotation, scale or opacity of the layer. When you paste this path to another layer or paste it as a motion path the new path will be created based on the center of the target layer or on the center of the composition when you are using a vector path as a motion path.
When you create your bezier path on the shape layer with the pen tool the position of the path is based on the center of the shape layer. That is why you'll see Ellipse 1>Transform>Position at some value other than 0, 0. If you select the Ellipse 1> Ellipse Path 1 path and convert it to a Bezier Path the Ellipse 1>Transform>Position property will be ignored so when you paste that path it will be based on the center of the shape layer's position and it will end up at the center of the shape layer. The Ellipse 1>Transform position (or rotation or scale) property will be ignored.
On the other hand, if you hold down the Alt/Option key when you draw your Shape Layer Ellipse (or any other parametric shape) a bezier path will be created based on the center of the shape layer so it will paste in the original location unless and just like a mask on a layer that has modified transform properties, the shape layer has been scaled, rotated or moved from the default layer transform properties of half comp width and height for position, 100% for scale and 0º for rotation.
Did you follow that? This is a behavior you have to learn to work with, not a bug. The only options I know of to get a copied shape layer path to end up in the exact same position as the original if the shape transform position properties are not at 0,0 is to copy those position properties and paste them to the transform properties of the new shape layer or set the shape>transform shape>position value to 0,0 to center the ellipse on the layer, then double click the Bezier Path to bring up the selection tools and move it back to the original position before you copy.
Does that make sense? If you would like to see a change in behavior please post a feature request. Personally, i'm used to the limitation and i never get caught by the behavior. I actually find it an advantage most of the time.
On your copy vector path to a motion path the same thing applies. The original vector path will be copied based on the layer's default position, scale and rotation. If you put a path on a scaled or repositioned layer and then copy it the motion path will be based on the comp center. Here's a quick tutorial that shows the basics of copying a path from a layer and getting it to move a camera in an efficient way. Enjoy: