We establish the existence and uniqueness of pathwise strong solutions to the stochastic 3D primitive equations with only horizontal viscosity and diffusivity driven by transport noise on a cylindrical domain $M=(-h,0)\times G$, $G\subset {\mathbb{R}}^2$ bounded and smooth, with the physical Dirichlet boundary conditions on the lateral part of the boundary. Compared to the deterministic case where the uniqueness of z-weak solutions holds in $L^2$, more regular initial data are necessary to establish uniqueness in the anisotropic space $H_z^1{L}_{xy}^2$ so that the existence of local pathwise solutions can be deduced from the Gyöngy-Krylov theorem. Global existence is established using the logarithmic Sobolev embedding, the stochastic Gronwall lemma and an iterated stopping time argument.