It is not clear what you are asking (what exactly do you want to generalize?), but let's start with a definition. Plurisubharmonic functions with logarithmic singularities can and have been be studied in complex manifolds of dimensions higher than $1$. In MR2133260
Rashkovskii, Alexander Sigurdsson, Ragnar
*Green functions with singularities along complex spaces*.
Internat. J. Math. 16 (2005), no. 4, 333–355 the authors consider a complex subspace $A$ of a connected complex manifold $X$ (not necessarily of codimension one) and $\mathcal{J}_A=(\mathcal{J}_A,x)_{x\in X}$-- the coherent sheaf of ideals in $\mathcal{O}(X)$ associated to $A$.
They define the class $\mathcal{F}_A$ as the set of all negative plurisubharmonic functions $u$ on $X$ such that for every point $a \in X$ there exist local generators $\psi_1,…,\psi_m$ for $\mathcal{J}_A$ near $a$ and a real constant $C$ (depending on $u$ and the generators) such that $u\leq \log\|\psi\|+C$ near $a$.

This cannot be applied to a compact $X$ (no non-constant plurisubharmonic functions!), but one still can develop somewhat analogous theory for so-called $\omega$-plurisubharmonic functions with respect to a closed real $(1,1)$-current $\omega$ on $X$ (when such functions exist). For the details I recommend
MR2203165 Guedj, Vincent; Zeriahi, Ahmed Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15 (2005), no. 4, 607–639. Of particular interest to you should be the beginning of Section 6, where $[\omega]$ is a the first Chern class of a holomorphic line bundle $L$ over $X$. The observation that there is a 1-to-1 correspondence between $\omega$-plurisubharmonic functions and positive singular metrics on $L$ should help you make sense of ``plurisubharmonic functions with logarithmic singularities along a divisor".