Deploying multiple beyond diagonal reconfigurable intelligent surfaces (BD-RISs) can potentially improve the communication performance thanks to inter-element connections of each BD-RIS and inter-surface cooperative beamforming gain among BD-RISs. However, a major issue for multi-BD-RISassisted communication lies in the channel estimation overhead - the channel coefficients associated with the off-diagonal elements in each BD-RIS's scattering matrix as well as those associated with the reflection links among BD-RISs have to be estimated. In this paper, we propose an efficient channel estimation framework for double-BD-RIS-assisted multi-user multipleinput multiple-output (MIMO) systems. Specifically, we reveal that high-dimensional cascaded channels are characterized by five low-dimensional matrices by exploiting channel correlation properties. Based on this novel observation, in the ideal noiseless case, we develop a channel estimation scheme to recover these matrices sequentially and characterize the closed-form overhead required for perfect estimation as a function of the numbers of users and each BD-RIS's elements and channel ranks, which is with the same order as that in double-diagonal-RIS-aided communication systems. This exciting result implies the superiority of cooperative BD-RIS-aided communication over the diagonal- RIS counterpart even when channel estimation overhead is considered. We further extend the proposed scheme to practical noisy scenarios and provide extensive numerical simulations to validate its effectiveness.