Soft gels, formed via the self-assembly of particulate organic materials, exhibit intricate multi-scale structures that provides them with flexibility and resilience when subjected to external stresses. This work combines molecular simulations and topological data analysis (TDA) to characterize the complex multi-scale structure of soft gels. Our TDA analysis focuses on the use of the Euler characteristic, which is an interpretable and computationally-scalable topological descriptor that is combined with filtration operations to obtain information on the geometric (local) and topological (global) structure of soft gels. We reduce the topological information obtained with TDA using principal component analysis (PCA) and show that this provides an informative low-dimensional representation of gel structure. We use the proposed computational framework to investigate the influence of gel preparation (e.g., quench rate, volume fraction) on soft gel structure and to explore dynamic deformations that emerge under oscillatory shear in various response regimes (linear, nonlinear, and flow). Our analysis identifies specific scales and extents at which hierarchical structures in soft gels are affected; moreover, correlations between structural deformations and mechanical phenomena (such as shear stiffening) are explored. In summary, we show that TDA facilitates the mathematical representation, quantification, and analysis of soft gel structures, extending traditional network analysis methods to capture both local and global organization.