Binary clusters (BCs) provide valuable observational constraints on the formation, early evolution, and dynamical interactions of star clusters. Their spatial and kinematic associations offer unique insights into the hierarchical star formation process and tidal interactions within the Galactic disk. We aim to identify and classify BCs using high-precision astrometric and kinematic data, and to investigate their physical properties, mutual gravitational interactions, and formation rates. We used a comprehensive star cluster catalog that contains 4084 high-quality clusters. Based on spatial and kinematic proximity, we identified 400 cluster pairs involving 686 unique clusters. These pairs were classified into three types: primordial BCs, systems formed through tidal capture or resonant trapping, and hyperbolic encounter pairs. For each system, we calculated the tidal factor to quantify the strength of mutual tidal interaction. Additionally, we constructed multi-cluster systems by identifying transitive connections among cluster pairs. Among the 400 identified cluster pairs, nearly 60.8% (243 pairs) are probably primordial BCs, exhibiting both similar ages and motions. This supports a scenario where they formed together in the same giant molecular cloud. We find that 82.5% of the cluster pairs have strong mutual tidal forces. In addition, 278 star clusters are identified as members of 82 multi-cluster systems, including 27 newly reported groups. Cross-matching with the literature confirms the recovery of previously reported systems and leads to the discovery of 268 new cluster pairs. In our sample, about 16.8% of star clusters are involved in some type of interaction with another cluster, and 9.94% of star clusters are likely born in primordial BCs. Our results provide a comprehensive, homogeneously identified sample of Galactic BCs. The high fraction of primordial BCs and their mutual tidal interaction suggest that cluster formation in pairs is a main outcome of star formation. This work offers new observational constraints on the formation and dynamical evolution of multiple star cluster systems.