Multi-dimensional earthquake isolation and mitigation device (MEIMD) is a newly developed structural vibration control device. First, the horizontal and vertical property tests are performed to study the influences of excitation frequency and displacement amplitude on the dynamic properties of the MEIMD. In order to accurately describe the nonlinear characteristics of the device caused by the complex viscoelasticity, an integrated mathematical model based on fractional-derivative equivalent standard solid model is then proposed. Next, the horizontal and vertical shaking table tests on a 1/5-scale three-story steel frame structure equipped with and without the MEIMDs are presented, respectively. Finally, a dynamic response analysis method considering the nonlinearity of the MEIMD is proposed to analyze the dynamic responses of the controlled structure. The analysis results show that the MEIMD can provide excellent horizontal isolation ability, and good vertical isolation performance can be achieved through selecting reasonable pre-pressure value of the springs. The proposed mathematical model and dynamic response analysis method can effectively describe the nonlinearity of the MEIMDs and the structure with MEIMDs.