The self-similarity of complex systems has been studied intensely across different domains due to its potential applications in system modeling, complexity analysis, etc., as well as for deep theoretical interest. Existing studies rely on scale transformations conceptualized over either a definite geometric structure of the system (very often realized as length-scale transformations) or purely temporal scale transformations. However, many physical and social systems are observed as temporal interactions among agents without any definitive geometry. Yet, one can imagine the existence of an underlying notion of distance as the interactions are mostly localized. Analysing only the time-scale transformations over such systems would uncover only a limited aspect of the complexity. In this work, we propose a novel technique of scale transformation that dissects temporal interaction networks under spatio-temporal scales, namely, flow scales. Upon experimenting with multiple social and biological interaction networks, we find that many of them possess a finite fractal dimension under flow-scale transformation. Finally, we relate the emergence of flow-scale self-similarity to the latent geometry of such networks. We observe strong evidence that justifies the assumption of an underlying, variable-curvature hyperbolic geometry that induces self-similarity of temporal interaction networks. Our work bears implications for modeling temporal interaction networks at different scales and uncovering their latent geometric structures.