The Gromov–Wasserstein (GW) distance is a widely adopted tool for comparing structured data, including shapes, point clouds, and networks, even when these reside in distinct ambient spaces. However, its applicability is limited by strict mass-conservation constraints and sensitivity to outliers, both of which hinder performance on real-world, imperfect datasets. To address these limitations, we consider the Conic Gromov–Wasserstein (CGW) distance, which accommodates unbalanced or partially observed structures. We introduce a novel formulation of CGW in terms of semi-couplings, extending its scope beyond classical metric spaces to encompass general graph and hypergraph representations. This framework enables the derivation of several fundamental properties, including scaling behaviour and robustness guarantees. Furthermore, empirical evaluations on real-world datasets demonstrate that the CGW distance remains computationally scalable, numerically stable, and interpretable even in the presence of structural and mass discrepancies.