The fluid–structure interaction (FSI) phenomenon in rectangular containers is a common and crucial physical process in nuclear engineering. Under seismic conditions, the interaction between liquid sloshing and container wall vibrations induces significant FSI effects, which directly impact the structural stability and seismic performance of rectangular containers. Therefore, studying FSI effects is essential for ensuring the seismic safety of nuclear engineering structures. Traditional liquid sloshing studies are mostly based on the pure fluid assumption, failing to fully account for FSI effects. This leads to significant deviations in structural response predictions under dynamic loads such as earthquakes. To address these challenges, this study develops a Fluid–Structure Interaction solver based on the Arbitrary Lagrangian–Eulerian (ALE) method to simulate liquid sloshing under FSI effects. Experimental data validation demonstrates that the proposed approach achieves high computational accuracy and reliability. This research provides an effective numerical tool for the seismic design and safety assessment of rectangular containers in nuclear engineering, offering significant academic and engineering value.