This article generalizes some geometric structures on warped product manifolds equipped with a Poisson structure to doubly warped products of pseudo-Riemannian manifolds equipped with a doubly warped Poisson structure. First, we introduce the notion of Poisson doubly warped product manifold (fB×bF,Π=μvΠBh+νhΠFv,g) and express the Levi-Civita contravariant connection, curvature and metacurvature of (fB×bF,Π,g) in terms of Levi-Civita connections, curvatures and metacurvatures of components (B,ΠB,gB) and (F,ΠF,gF). We also study compatibility conditions related to the Poisson structure Π and the contravariant metric g on fB×bF, so that the compatibility conditions on (B,ΠB,gB) and (F,ΠF,gF) remain consistent in the Poisson doubly warped product manifold (fB×bF,Π,g).