To address the performance degradation of fully differential frequency-modulated Hemispherical Resonator Gyroscope (HRG) caused by assembly errors, this paper proposes a synchronous identification and compensation method for assembly errors based on nonlinear optimization from a signal processing perspective. First, a systematic correlation model between the assembly attitude errors and channel coupling errors under the time-division multiplexing control scheme is established, and the influence mechanisms of installation tilt and installation eccentricity on channel coupling errors are analyzed. On this basis, a dynamic output model of the HRG incorporating channel coupling errors is developed, revealing the coupling effect between the carrier rotation rate and the harmonic components of the gyroscope output. Finally, a synchronous identification and compensation method for assembly errors based on nonlinear optimization is proposed. Experimental results demonstrate that after compensating for assembly errors, the scale factor nonlinearity and circumferential drift stability of the gyroscope are reduced by 91.97% and 51.25%, respectively, reaching only 6.94×10^-7 and 0.398 (°)/h. This method significantly improves the gyroscope's performance in both dynamic and static environments, providing theoretical support for the design and optimization of the high-precision HRG.