My question comes as a natural follow-up of the previous one which concerned symplectic manifolds: if $(M, P)$ is a Poisson manifold, what embedding theorems are there into some target space (I am looking mostly at embedding into Euclidean or projective spaces, real or complex). Having learned my lesson from the previous question, I am willing to be much more modest: since embeddings might be too much, are there just Poisson maps between $(M,P)$ and $\Bbb R^n$ or some projective space (with their standard non-degenerate Poisson structures)?

Not in this form. Poisson maps are rank decreasing. If you have a Poisson map $f$ from $(M,\pi_1)$ to $(M,\pi_2)$ then ${\mathrm rank}\,\pi_1(x)\ge {\mathrm rank}\,\pi_2(f(x))$. Therefore if the target manifold is symplectic then $f$ should be a submersion.

Even $\mathbb R^2\hookrightarrow \mathbb R^4$ with respect to the standard Poisson bracket is not a Poisson map.

Embedding of a Poisson manifold, meaning Poisson submanifolds, implies being a union of symplectic leaves. A "universal" Poisson manifold containing all possible manifolds as Poisson submanifold is a little bit weird to imagine.

The point here is that symplectic maps and Poisson maps are quite different in general.