There are [lecture notes by Roubtsov-Suchánek][1] on Poisson algebras, where the subject is rendered very algebraically. I believe section 4 has an answer to your question, especially Proposition 4.1., almost by definition.

Suppose that our Poisson structure comes from a non-degenerate closed skew-form $\omega$.
The map $\varphi :H\mapsto -X_H$ is a homomorphism of Lie algebras and the center is its kernel (this is where the space of functions gets its structure of a Lie algebra). But this map is nothing more than a composition of the differential of a function with the isomorphism induced by $\omega$, hence the differential of an element of the kernel should be identically zero.

  [1]: https://arxiv.org/abs/2305.03578v1