The damping performance evaluation for electromechanical oscillations in power systems is crucial for the stable operation of modern power systems. In this paper, the connection between two commonly-used damping performance evaluation methods, i.e., the damping torque analysis (DTA) and energy flow analysis (EFA), are systematically examined and revealed for the better understanding of the oscillatory damping mechanism. First, a concept of the aggregated damping torque coefficient is proposed and derived based on DTA of multi-machine power systems, which can characterize the integration effect of the damping contribution from the whole power system. Then, the pre-processing of measurements at the terminal of a local generator is conducted for EFA, and a concept of the frequency-decomposed energy attenuation coefficient is defined to screen the damping contribution with respect to the interested frequency. On this basis, the frequency spectrum analysis of the energy attenuation coefficient is employed to rigorously prove that the results of DTA and EFA are essentially equivalent, which is valid for arbitrary types of synchronous generator models in multi-machine power systems. Additionally, the consistency between the aggregated damping torque coefficient and frequency-decomposed energy attenuation coefficient is further verified by the numerical calculation in case studies. The relationship between the proposed coefficients and the eigenvalue (or damping ratio) is finally revealed, which consolidates the application of the proposed concepts in the damping performance evaluation.