What is the physically reasonable equivalent of mass for the electric permettivity?

When dealing with physical problems in which the mass is involved, the function density of mass (call it $rho(x)$) is very useful in computations to study the mathematical modellings.
If one knows the mass distribution, the mass of an object P occupying a region in the space (call it $Omega subset mathbb{R}^3$) is given by the integral over the region of its density of mass, i.e.
$$operatorname{Mass}(P) = int_Omega rho(x) dx.$$

Now, if in certain kind of problems, for example arising from Newton’s laws of motion, a fundamental concept is mass and consequently density of mass, when dealing with issues involving Maxwell equations, one of the fundamental properties of the objects under consideration is the electric permittivity $varepsilon$ of a material, which is usually a function that to every point of the space in which the material is situated gives back a $3 times 3$ matrix (it can be reduced to a scalar function in case the material is isotropic).

Going back, the mass distribution is something like the infinitesimal concentration of the mass in a small region, meaning that for example it can be viewed as the following limit

$$rho(x)=lim_{r to 0} frac{operatorname{Mass}(B_r(x))}{operatorname{meas}(B_r(x))},$$
where $B_r(x)$ is the ball of radius $r$ centered in $x$.

Is there an equivalent view for the electric permittivity? Is it the infinitesimal spatial distribution of a certain quantity, a sort of "electric mass" or "electric energy" of the object (here the notion of capacitance comes into mind)? In that case, what would be that quantity and how can one obtain it if he is given $varepsilon(x)$ in every point $x$ of the space? Would any of these integrals
$$int_Omega varepsilon(x) dx, quad int_Omega |varepsilon(x)|^2 dx,$$
or something similar involving $varepsilon(x)$, make any sense in terms of physical interpretation?

I understand that the electric permittivity is something like the index of how well can a material acquire "affinity" with an electric field flowing through it, in the sense that the electric field would flow and polarize the material better or worse depending on the permittivity index. But I still don’t fully grasp the concept of permittivity, what does it embody.

If any of you has any idea or intuition, it would be greatly appreciated. Moreover, a parallel question could be raised for the magnetic part of the whole topic, regarding the magnetic permeability $mu(x)$.