Reconfigurable intelligent surface (RIS) is a promising technique for millimeter wave (mmWave) positioning systems. In this paper, we consider multiple mobile users (MUs) positioning problem in the multiple-input multiple-output (MIMO) time-division duplex (TDD) mmWave systems aided by the RIS. We derive the expression for the space-time channel response vector (STCRV) as a novel type of fingerprint. The STCRV consists of the multipath channel characteristics, e.g., time delay and angle of arrival (AOA), which is related to the position of the MU. By using the STCRV as input, we propose a novel residual convolution network regression (RCNR) learning algorithm to output the estimated three-dimensional (3D) position of the MU. Specifically, the RCNR learninng algorithm includes a data processing block to process the input STCRV, a normal convolution block to extract the features of STCRV, four residual convolution blocks to further extract the features and protect the integrity of the features, and a regression block to estimate the 3D position. Extensive simulation results are also presented to demonstrate that the proposed RCNR learning algorithm outperforms the traditional convolution neural network (CNN).