  We extend to any dimension the quantitative fourth moment theorem on the
Poisson setting, recently proved by C. D\"obler and G. Peccati (2017). In
particular, by adapting the exchangeable pairs couplings construction
introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove
our results under the weakest possible assumption of finite fourth moments.
This yields a Peccati-Tudor type theorem, as well as an optimal improvement in
the univariate case. Finally, a transfer principle "from-Poisson-to-Gaussian"
is derived, which is closely related to the universality phenomenon for
homogeneous multilinear sums.
