The accurate and robust simulation of compressible flows remains a major challenge in computational fluid dynamics, particularly in the presence of shocks, strong nonlinear wave interactions, and multi-scale effects. Although high-order discontinuous Galerkin (DG) methods provide high accuracy on unstructured meshes, ensuring robustness and preserving fundamental geometric and energetic structures for nonlinear systems such as the compressible Euler equations remain challenging. These limitations call for discretization frameworks that respect the intrinsic energy-based structure of the governing equations. Port-Hamiltonian formulations provide a geometric framework for the two- and three-dimensional compressible Euler equations \cite{vanDerSchaft2002}, and recent structure-preserving DG methods have been developed for linear port-Hamiltonian systems \cite{Cheng2025PHDG}. However, a unified DG framework extending these ideas to nonlinear compressible flows with energy-consistent inter-element coupling is still lacking. In this work, we propose a unified structure-preserving DG framework for nonlinear compressible flows based on a port-Hamiltonian representation of the Euler equations. The computational domain is partitioned into non-overlapping elements, and a weak Dirac structure is formulated on the resulting tessellated domain using broken Sobolev spaces within an input–state–output framework. This construction ensures that power-conserving interconnections between neighboring elements remain Dirac structures, so that the Hamiltonian of the global system equals the sum of the Hamiltonians of the individual elements. As a consequence, energy exchange in the semi-discrete system occurs only through the external boundary of the computational domain, while all inter-element interfaces correspond to power-conserving internal interconnections. The induced weak Dirac structure leads to a Poisson bracket satisfying the geometric requirements of a Hamiltonian formulation. By approximating the broken Sobolev spaces with broken polynomial spaces, we obtain a structure-preserving DG semi-discretization that guarantees internal energy conservation and boundary-controlled energy transfer. The proposed framework provides a systematic foundation for high-order DG schemes with improved robustness and physical consistency, and its performance is demonstrated through numerical experiments for standard two- and three-dimensional compressible Euler benchmark problems.