Identifiability is a central issue in blind source separation (BSS), determining whether latent sources can be uniquely recovered from observed mixtures. Classical approaches address identifiability either by exploiting source non-Gaussianity via higher-order statistics (HOS) or by enriching the observation structure through temporal, spatial, or multi-channel diversity using second-order statistics (SOS), and these routes are often regarded as fundamentally different. In this paper, we revisit identifiability in BSS from a structural perspective, interpreting it as constraint-induced reduction of residual ambiguity in the mixing model. Within this framework, the observation mechanism is viewed broadly to include both input-side statistical constraints and output-side observation structures. HOS-based and SOS-based approaches are then unified as mechanisms of stabilizer shrinkage, in which observation-induced constraints reduce an initially continuous ambiguity to a finite residual one. To connect this structural viewpoint with finite-sample regimes, we introduce a Jacobian-based sensitivity probe as a numerical diagnostic of local identifiability. Numerical experiments show that increasing non-Gaussianity or observation diversity suppresses the same residual symmetry, revealing a structural trade-off between source statistics and observation design. These results provide a unified interpretation of classical BSS methods and clarify how observation constraints govern identifiability.