Understanding how nonlinear dynamical systems (e.g., artificial neural networks and neural circuits) process information requires comparing their underlying dynamics at scale, across diverse architectures and large neural recordings. While many similarity metrics exist, current approaches fall short for large-scale comparisons. Geometric methods are computationally efficient but fail to capture governing dynamics, limiting their accuracy. In contrast, traditional dynamical similarity methods are faithful to system dynamics but are often computationally prohibitive. We bridge this gap by combining the efficiency of geometric approaches with the fidelity of dynamical methods. We introduce fast dynamical similarity analysis (fastDSA), a computationally efficient and accurate metric for measuring (dis)similarity between nonlinear dynamical systems. FastDSA leverages modern computational tools, including random matrix theory to determine optimal system rank, novel optimization pipelines for aligning system flow fields, and Koopman embeddings. Across benchmark nonlinear systems and recurrent network models, fastDSA is robust to arbitrary coordinate choices while remaining sensitive to meaningful dynamical differences, capturing variations in system evolution that geometric methods may miss and traditional methods detect only at high computational cost. To our knowledge, fastDSA is the fastest method that retains accuracy in comparing nonlinear dynamical systems. It enables scalable, statistical analyses across diverse systems, significantly expanding the practical applicability of dynamical similarity analysis.