The function $\psi:=\log(fh^{-1})$ satisfies $\partial\bar\partial \psi=0$, because $ \partial\bar\partial\log f=\partial\bar\partial\log h =\omega$.
Such functions are called pluriharmonic. Locally a pluriharmonic function is a real part of a holomorphic function, by Poincare-Dolbeault-Grothendieck lemma.
This fact is true globally, because any real-valued holomorphic function vanishes, and therefore the local holomorphic functions can be glued together. Then, $h$ is unique (up to a constant multiplier) if and only if the manifold has no non-constant global holomorphic functions.