Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think it helps).



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> > Is it true that if $\frac{\partial^2}{\partial z\_i\partial \overline{z\_j}}\log \|f(z)\|=\frac{\partial^2}{\partial z\_i\partial \overline{z\_j}}\log \|g(z)\|$, for all $i,j\le n$, then there is a holomorphic function $h:G\to\mathbb{C}$ and an isometry $U:H\to H$, such that $g(z)=h(z)Uf(z)$?
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**Motivation.** Consider the canonical quotient $p:H\to PH$, where the latter is the projective space over $H$. Suppose $\varphi:G\to G$ is a biholomorphism, such that $f$ and $g=f\circ\varphi$ satisfy that condition. Then $pf\varphi(pf)^{-1}$ is an isometry of $pf(G)$ with respect to the Fubini-Study metric on $PH$. I wonder, if this isometry can be lifted to an isometry $H$, i.e. if there is an isometry $U:H\to H$, such that $pUf=pf\varphi$. Finally the last condition means that there is a holomorphic function $h:G\to\mathbb{C}$, such that $h(z)Uf(z)=f(\varphi(z))$.



Derivatives of norm of vector-valued holomorphic functions
Yes, this is true, and this is called the Calabi rigidity, though it was proved 
(for finite dimensional Hilbert space)
long before Calabi, see, for example, Polya-Szego, revised edition, part IV, problem 207, where there is a reference on the original paper.
Calabi's paper is: Isometric imbedding of complex manifolds, Ann. Math. 58
(1953), 1–23.


Another source is Theorem 2 in <https://arxiv.org/pdf/math/0007030.pdf>.


