Is zero-point vibrational energy an intensive or extensive property?
Mathematically, an extensive property $f$ is one for which $f(\lambda x,\lambda y,z) = \lambda f(x,y,z)$ for extensive variables $x$, $y$, and intensive variable $z$. For example, the Helmholtz free energy $A(T,\lambda V,\lambda N) = \lambda A(T,V,N)$ is extensive.

Following this definition, the zero-point energy $$E_\text{ZPVE}(\{\omega_c\},\lambda n) = \lambda E_\text{ZPVE}(\{\omega_c\},n),$$ where $n$ is the number of normal modes, is extensive in the thermodynamic limit $N \to \infty$.

The polarizability should be able to be treated similarly.



It bears noting that a lot of macroscopic extensive properties are not extensive in the microscopic regime. For example, if we consider two identical systems, each with an energy $U$, then the combined system has energy $2U$ plus an additional energetic term $U_\text{sys-sys}$, accounting for superficial energetic interactions between the two subsystems. The surface, and hence the superficial energy term, scales as one dimension removed of the volume, and likewise vanishes in the thermodynamic limit.

Furthermore, it's perfectly valid to have properties that are neither extensive nor intensive, like the number of microstates that a system possesses.