This paper focuses on a mean-field optimal stopping problem with non-Markov dynamics and common noise, inspired by (Talbi, Touzi, and Zhang 2022). The goal is to establish the limit theory and demonstrate the equivalence of the value functions between weak and strong formulations. The difference between the strong and weak formulations lies in the source of randomness determining the stopping time on a canonical space. In the strong formulation, the randomness of the stopping time originates from Brownian motions. In contrast, this may not necessarily be the case in the weak formulation. Additionally, a (H)-Hypothesis-type condition is introduced to guarantee the equivalence of the value functions. The limit theory encompasses the convergence of the value functions and solutions of the large population optimal stopping problem towards those of the mean-field limit, and it shows that every solution of the mean field optimal stopping problem can be approximated by solutions of the large population optimal stopping problem.