How is the centre of mass of a Positronium associated with that of an electron and positron? How is the centre of mass of a Positronium likely associated with that of an electron and positron?
 A: Positronium is a quantum mechanical entity. It is a bound state in the  potential create between two elementary "particles". Even though the wiki article draws a nice picture of he electron and positron orbiting about their center of mass this is a representation as if they are classical particles . They are not.
The electron and positron are not in orbits, but in orbitals. The difference between an orbit and an orbital is the difference between classical mechanics and quantum mechanics. In the QM case the orbital is a locus in space where the probability of finding an electron (positron) in a specific (x,y,z) is given.
Macroscopically the positronium can be located by the center of mass, but macroscopically also one cannot see the electron and positron as separate particles. It is one classical particle with the mass of two electrons,  and its center of mass is imilar to the center of mass of any atom: within classical dimensions it coincides with its position as a point.
Microscopically:  the wiki picture can be seen as  defining a center of mass statistically, as momentum and angular momentum have to be conserved.
There is no measurable meaning to it  since if we manage to measure the x,y,z of the electron it means that there is no longer a positronium but two different elementary particles. From the values of the electron  using the conservation laws we can define a center of mass  that existed before our measurement. 
