No, current density is not a property of the medium.
It's not a property of the medium itself, but it can be related to some property of the medium, like resistivity $\sigma$, and to other electromagnetic quantities, like the electric $\mathbf{e}$ field at some point of the medium $\mathbf{r}$, as an example with Ohm's law,
\begin{equation}
\mathbf{e}(\mathbf{r}) = \sigma(\mathbf{r}) \, \mathbf{j}(\mathbf{r}) \ .
\end{equation}
This is the local (differential) form of the "most famous" $\Delta v = R i$ for a linear resistor in circuit theory.
Even though $\mathbf{j}$ is not a property of the medium, it should be clear that the medium, through its properties, could influence the current flowing in it. As an example, the same electric field applied to two different materials with resistivity $\sigma_1$, $\sigma_2$ would produce currents that are inversely proportional to resistivity of the materials,
$$ \mathbf{j}_1 = \frac{\mathbf{e}}{\sigma_1} = \frac{\sigma_2}{\sigma_1} \, \mathbf{j}_2 \ .$$
Current density in porous electrode. I'm not an expert of the field, but I guess that the current density is higher in porous media, since you're considering the external surface of the electrode in the ratio $\frac{current}{area}$, while porosity implies a larger useful surface w.r.t. flat "classical" electrodes. It's not (only) a matter of the material itself, but the useful surface given the same exterior surface.
