Motivated by the wide presence of multilayer networks in both natural and human-made systems, within a random matrix theory (RMT) approach, in this study we compute eigenfunction and spectral properties of multilayer directed random networks (MDRNs) in two setups composed by $M$ layers of size $N$: A line and a complete graph (node-aligned multiplex network). First, we numerically demonstrate that the normalized localization length $β$ of the eigenfunctions of MDRNs follows a simple scaling law given by $β=x^*/(1+x^*)$, with $x^*\propto (b_{\rm eff}^2/L)^δ$, $δ\sim 1$ and $b_{\rm eff}$ being the effective bandwidth of the adjacency matrix of the network of size $L=M\times N$. Here, $b_{\rm eff}$ incorporates both intra- and inter-layer edges. Then, we show that other eigenfunction and spectral RMT measures (the inverse participation ratio of eigenfunctions, the ratio between nearest- and next-to-nearest-neighbor eigenvalue distances, and the ratio between consecutive singular-value spacings) of MDRNs also scale with $x^*$.