Existing studies have confirmed that fluid–structure interactions (FSIs) between flexible structures and fluids can enhance heat transfer in enclosed and ventilated cavities, with relevant characteristics affected by structural elasticity, flow parameters, and other factors. However, current integer-order models fail to account for the viscoelastic memory effect of time-fractional Maxwell fluids and lack the application of fractional derivatives, making it difficult to accurately characterize the dynamic flow–deformation interaction. Taking the square cavity flow with flexible fins as the research object, and based on the advantages of Caputo fractional derivatives—including compatibility with physical initial conditions, simplicity of numerical discretization, and a superior ability to fit long-term memory characteristics—this paper introduces Caputo fractional Maxwell shear stress into the constitutive relation and incorporates time-fractional derivatives into the momentum and energy equations, respectively, to establish the governing equations, thereby constructing an FSI model of the fluid and flexible fins. Numerical simulations comparing the coupling results of integer-order and fractional-order fluids show that the fractional order (α) directly regulates the intensity of viscous memory; analysis of differences in velocity distribution, vortex structure, and wall shear stress reveals that the fractional-order model exhibits a more lagging flow response. The research indicates that fractional derivatives are a key tool for characterizing the long-term memory properties of Maxwell fluids, which can effectively improve the physical authenticity of FSI models and provide reliable theoretical support for engineering applications such as micro flexible fluid devices and fluid transportation.