This goes back to the ambiguity of the definition of orbit. If I have a system with two points involved in some sort of rotational motion, then there isn't a "best choice" for which point should be considered the origin of my coordinate system.
If you mean exactly 2 abstract points in Euclidean geometry then sure. 3D objects are not points which is the first issue. Relatively also cares about accelerating or rotating reference frames which applies to both point masses.
The center of mass in a two body system is a nice inertial reference frame which simplifies calculations.
