The dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD) are used to analyze the coherent structures of turbulent flow around vibrating isolated and piggyback cylinders configurations subjected to a uniform flow at a laminar Reynolds number (Re = 200) and a upper transition Reynolds number (Re = 3.6 × 106). Numerical simulations using two-dimensional unsteady Reynolds-averaged Navier–Stokes (URANS) approach with the k − ω shear stress transport turbulence model are used to obtain the flow fields snapshots for the analysis. The wake flows behind the cylinders are decomposed into energy optimal modes (POD modes) and dynamical relevant modes (DMD modes). A reduced-order model (ROM) for the flow is built based on the modal analysis. A comparison of POD and DMD is performed to characterize their special features. The present study provides new insights into the flow physics of fluid–structure interaction problem of two coupled cylinders. The characteristic vortex shedding frequencies and their harmonics are identified by DMD modes in all the investigated configurations. It is observed that for single cylinder configurations, the most energetic and the most dynamically important mode is associated with the fundamental shedding frequency. For the stationary piggyback configuration, the gap flow between the cylinders appears to be a dominant flow feature as evidenced by the leading DMD modes. The cylinder vibration increases significantly number of modes necessary to obtain a ROM at the given level of accuracy compared to respective stationary configurations.