From your vague dismissal of David Lehavi's answer, it seems what you want is a distance between embeddings of ellipsoids.  So would the following idea be right for what you want?  Given two embedded ellipsoids in $\mathbb{R}^3$, translate and rotate $\mathbb{R}^3$ so that $C_A$ is $\vec{0}$ and $M_A$ is diagonal with eigenvalues ordered by size.  What we're doing here is putting ellispoid A in a standard position.

We can take the transformation that took A to standard position and apply it to ellipsoid B which now moves to some new position.  Ellipsoid B is now characterized by $C_B'$ and $M_B'$ ($C_B' = C_B-C_A$, $M_B'= O^T M_B O$, where $O$ is the rotation diagonalizing $M_A$ as described previously)

One can now assign a "distance" between A and B which depends only on the eigenvalues of $M_A$, the vector $C_B'$ and the matrix $M_B'$.  By moving your ellipsoids together to this "standard position" we've quotiented out the space of "pairs of embedded ellipsoids" by the action of rigid transformations of $\mathbb{R}^3$, as presumably you want your distance function to be invariant under this group.

There are many distances one can assign to ellipsoids in standard position, depending on how you weight "shape" vs. "distance" vs. "orientation", etc. - what you choose is going to depend heavily on what you want.  So please do give your needs a little more thought.