While it does not seem to be possible to copy a shape without selecting it, it is possible to duplicate a shape without selecting it (which was my reason for wanting to copy it in the first place). The code below gives me what I was looking for:
Sub createShape()
Set myshape = ActiveDocument.Shapes.AddShape(msoShapeRectangle, 100, 100, 100, 100)
Set anothershape = myshape.Duplicate
End Sub