Reconfigurable Intelligent Surfaces (RIS) represent a transformative technology for sixth-generation (6G) wireless communications, but it suffers from a significant limitation, namely the double-fading attenuation. Active RIS has emerged as a promising solution, effectively mitigating the attenuation issues associated with conventional RIS-assisted systems. However, the current academic work on active RIS focuses on the system-level optimization of active RIS, often overlooking the development of models that are compatible with its electromagnetic (EM) and physical properties. The challenge of constructing realistic, EM-compliant models for active RIS-assisted communication, as well as understanding their implications on system-level optimization, remains an open research area. To tackle these problems, in this paper we develop a novel EM-compliant model with mutual coupling (MC) for active RIS-assisted wireless systems by integrating the developed scattering-parameter ($S$-parameter) based active RIS framework with multiport network theory, which facilitates system-level analysis and optimization. To evaluate the performance of the EM-compliant active RIS model, we design the joint optimization scheme based on the transmit beamforming at the transmitter and the reflection coefficient at the active RIS to maximize the achievable rate of EM-compliant active RIS-assisted MIMO system. To tackle the inherent non-convexity of this problem, we employ the Sherman-Morrison inversion and Neumann series (SMaN)-based alternating optimization (AO) algorithm. Simulation results verified that EM property (i.e., MC effect) is an indispensable factor in the optimization process of MIMO systems. Neglecting this effect introduces a substantial performance gap, highlighting its significance in the more pronounced the MC effect is, the greater the gap in achievable rates.