In this work, we investigate the use of backscattered mm-wave radio signals for the joint tracking and recognition of identities of humans as they move within indoor environments. Previous research has considered a single-person identification problem, while the multi-person case was only addressed in an offline fashion through the superposition of multiple single-person signals. In contrast, we build a system that effectively works with multiple persons concurrently sharing and freely moving within the same indoor space. This leads to a complicated setting, which requires one to deal with the randomness and complexity of the resulting (composite) backscattered signal. Our solution features a novel signal processing pipeline: first, the signal is filtered to remove artifacts, reflections and random noise that do not originate from humans. A following density-based classification algorithm is executed to separate the Doppler signatures of different users. The last two blocks are trajectory tracking and user identification, respectively based on Kalman filters and deep neural networks. Our results demonstrate that the integration of these two processing stages is critical towards achieving robustness and accuracy in multi-user settings. The proposed system is tested both on a single-target public dataset, for which it outperforms state-of-the-art techniques, and on our own measurements, obtained with a 77 GHz radar on multiple subjects simultaneously moving in an indoor environment. The system works in an online fashion, permitting the continuous identification of multiple subjects with accuracies up to 98%, e.g., with four subjects sharing the same physical space.