Motivated by the need to provide differentiated quality-of-service (QoS) in grant-free uplink transmissions in 5G networks and beyond, we extend the probabilistic analysis of coded Poisson receivers (CPR) to the setting with multiple classes of users and receivers. For such a CPR system, we prove (under certain technical conditions) that there is a region, called the stability region in this paper. Each transmitted packet can be successfully received with probability 1 when the offered load to the system is within the stability region. On the other hand, if the offered load is outside the stability region, there is a nonzero probability that a packet will fail to be received. We then extend the stability region to the $\epsilon$-stability region for CPR systems with decoding errors. We also demonstrate the capability of providing differentiated QoS in such CPR systems by comparing the stability regions under various parameter settings.