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Recently universal dynamic scaling is observed in several systems, which exhibit a spatiotemporal self-similar scaling behavior, analogous to the spatial scaling near phase transitions. The latter arises from the emergent continuous scaling symmetry due to the divergent correlation length. Motivated by this, we investigate the relation between the scaling dynamics and continuous scaling symmetry. We derive a theorem that the scaling invariance of the quenched Hamiltonian and the initial density matrix can lead to the universal dynamic scaling in quench dynamics. It is demonstrated both in the two-body problem analytically and in the many-body problem numerically. For the latter one, we calculate the dynamics of quantum gases quenched from noninteracting to finite interaction in the framework of non-equilibrium high-temperature virial expansion. A dynamic scaling of the momentum distribution appears in certain momentum-time windows at unitarity as well as in the weak interacting limit. Remarkably, this universal scaling dynamics persists approximately with smaller scaling exponents even if the scaling symmetry is fairly broken. Our findings may offer a new perspective to interpret the related experiments. We also calculate the Contact dynamics in the BEC-BCS crossover. Surprisingly, the half-way time displays a maximum near unitarity while some damping oscillations occur on the BEC side due to the dimer state, which can be used to detect possible two-body bound states in experiments.