Unravelling the coupling between river network structure and hydrological fluxes is essential for understanding basin-scale dynamics. While traditional ordering methods describe macro-scale patterns, they often obscure local functional variations. This study quantifies the universal hydrological patterns of global river networks by integrating the classical Horton–Strahler framework with a hierarchical pyramid decomposition technique. Leveraging the global HydroATLAS dataset, we analysed 228 representative basins spanning diverse hydro-climatic regimes. We extracted rigorously defined network attributes and hydrological fluxes to examine the scaling behaviours of fundamental structural components, defined here as basic units. Our results reveal a striking topological invariance in hydrological characteristics across both varying spatial scales and distinct geographic regions. Specifically, the runoff and discharge ratios of these basic units maintain robust statistical consistency regardless of basin size or climatic conditions ranging from humid to arid. This suggests that network topology functions as a dominant physical control, effectively acting as a low-pass filter that dampens high-frequency climatic variability to produce unified global scaling laws. These findings advance the theoretical understanding of fractal river networks. Furthermore, they open new avenues for prospective research, including the integration of these physics-informed topological priors into next-generation Earth system models to improve discharge predictions and water resource modelling in ungauged basins.