If $\varepsilon$ or $\mu$ are tensors (read, matrices), then so is $c_m$: $$ \overbrace{\varepsilon}^\mathrm{matrix} \underbrace{\mu}_\mathrm{matrix}=c_m^{-2}\ \leftarrow\ \text{matrix as well} $$

In other words, if the permeability and/or permittivity are matrices, then the speed of light is a matrix as well. In this case, the $\square^{\color{red}{-1}}$ is understood in the sense of matrix inverse.

In the coordiante system where $c_m$ is diagonal, we have $$ c_m=\begin{pmatrix} c_1&\cdot&\cdot\\\cdot&c_2&\cdot\\\cdot&\cdot&c_3\end{pmatrix} $$ where $c_i$ is the speed of light along the $x_i$ axis.

If $\varepsilon$ or $\mu$ are tensors (read, matrices), then so is $c_m$: $$ \overbrace{\varepsilon}^\mathrm{matrix} \underbrace{\mu}_\mathrm{matrix}=c_m^{-2}\ \leftarrow\ \text{matrix as well} $$

In other words, if the permeability and/or permittivity are matrices, then the speed of light is a matrix as well. In this case, the $\_^{\color{red}{-1}}$ is understood in the sense of matrix inverse.

In the coordiante system where $c_m$ is diagonal, we have $$ c_m=\begin{pmatrix} c_1&\cdot&\cdot\\\cdot&c_2&\cdot\\\cdot&\cdot&c_3\end{pmatrix} $$ where $c_i$ is the speed of light along the $x_i$ axis.

So yes: the speed of light in a medium can be direction-dependent; see Birefringence.