Summary: This article extends the tensor network Kalman filter to matrix outputs with an application in recursive identification of discrete-time nonlinear Multiple-Input Multiple-Output (MIMO) Volterra systems. This extension completely supersedes previous work, where only \(l\) scalar outputs were considered. The Kalman tensor equations are modified to accommodate for matrix outputs and their implementation using tensor networks is discussed. The MIMO Volterra system identification application requires the conversion of the output model matrix with a row-wise Kronecker product structure into its corresponding tensor network, for which we propose an efficient algorithm. Numerical experiments demonstrate both the efficacy of the proposed matrix conversion algorithm and the improved convergence of the Volterra kernel estimates when using matrix outputs.