In this work, we study the mean field Schrödinger problem from a purely probabilistic point of view by exploiting its connection to stochastic control theory for McKean-Vlasov diffusions. Our main result shows that the mean field Schrödinger problem arises as the limit of ``standard'' Schrödinger problems over interacting particles. Due to the stochastic maximum principle and a suitable penalization procedure, the result follows as a consequence of novel (quantitative) propagation of chaos results for forward-backwards particle systems. The approach described in the paper seems flexible enough to address other questions in the theory. For instance, our stochastic control technique further allows us to solve the mean field Schrödinger problem and characterize its solution, the mean field Schrödinger bridge, by a forward-backward planning equation.