Channel parameter recovery is critical for the next-generation reconfigurable intelligent surface (RIS)-empowered communications and sensing. Tensor-based mechanisms are particularly effective, inherently capturing the multi-dimensional nature of wireless channels. However, existing studies assume either a line-of-sight (LOS) scenario or a blocked TX-RX channel. This paper solves a novel problem: tensor-based channel parameter estimation for active RIS-aided multiple-antenna broadband connections in fully multipath environments with the TX-RX link. System settings are customized to construct a fifth-order canonical polyadic (CP) signal tensor that matches the five-dimensional channel. Four tensor factors contain redundant columns, rendering the classical Kruskal's condition for decomposition uniqueness unsatisfied. The fifth-order Vandermonde structured CP decomposition (VSCPD) is developed to address this challenge, making the tensor factorization problem solvable using only linear algebra and offering a relaxed general uniqueness condition. With VSCPD as a perfect decoupling scheme, a sequential triple-stage channel estimation algorithm is proposed based on one-dimensional parameter estimation. The first stage enables multipath identification and algebraic coarse estimation. The following two stages offer optional successive refinements at the cost of increased complexity. The closed-form Cramer-Rao lower bound (CRLB) is derived to assess the estimation performance. Herein, the noise covariance matrix depends on multipath parameters in our active-RIS scenario. Numerical results are provided to verify the effectiveness of proposed algorithms under various evaluation metrics. Our results also show that active RIS can significantly improve channel estimation performance compared to passive RIS.