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The retention of copper (Cu) in saline–alkali soil (SAS) during long-term application of biogas residue (BR) with a high concentration of Cu raises concerns. In this work, the fate of Cu was detected using adsorption isotherms, scanning electron microscope—energy dispersive spectrometer, Fourier transform infrared spectrometer, X-ray diffraction, isothermal titration calorimetry, X-ray photoelectron spectroscopy, and microzone X-ray fluorescence spectrometer. The results showed that the main groups for Cu adsorption by SAS and BR were carboxyl, hydroxyl, amide and amine. The adsorption of Cu by the carboxyl group was entropy–enthalpy co-driven (|ΔH| < |TΔS|, ΔH < 0). The adsorption of Cu by the amine group was entropy-driven (|ΔH| > |TΔS|, ΔH > 0). The adsorption of Cu on the SAS and BR was achieved by organic matter rather than minerals. The degradation of BR in the SAS increases the content of Cu adsorption groups such as carboxyl and amine groups, and Cu was adsorbed on the surface or inside SAS through organic groups. This study provides further theoretical support for the application of BR in SAS.