The damping asymmetry error is a prominent factor influencing the static performance of fully symmetrical Micro-Electro-Mechanical Systems (MEMS) vibratory gyroscopes. To eliminate this error and enhance the gyroscope's static performance, this paper introduces a method for real-time identification and automatic matching of the damping asymmetry error in rate mode. By leveraging the vibration characteristics of the fully symmetrical MEMS vibratory gyroscope in rate mode, dedicated online test methods are devised to determine the quality factors for both the drive and sense modes, enabling online identification of the damping asymmetry error. Furthermore, an automatic damping-matching system is devised utilizing the mechanical-thermal dissipation mechanism of a resistive damper. The effectiveness of this proposed method is validated through simulations and experiments conducted on a MEMS disk resonator gyroscope (DRG). The experimental results show an accuracy within 4% for quality factor identification, while automatic damping matching effectively reduces the damping asymmetry error by 87.64%. Employing this automatic damping-matching method successfully diminishes the zero-input bias caused by damping asymmetry error in the MEMS DRG from 0.1067°/s to 0.0249°/s and improves bias instability from 0.3985°/s to 0.1878°/s.