In this paper, we demonstrate the impact of channel aging on the spectral- and energy-efficiency performance of the multi-pair two-way massive MIMO relay (MTW-MMR) systems, where massive antennas are deployed at the relay and the amplify-and-forward (AF) protocol is operated. By assuming the maximum-ratio combining/maximum-ratio transmission (MRC/MRT) or zero-forcing reception/ zero-forcing transmission (ZFR/ZFT) relaying processing, we analyze the spectral efficiency (SE) and energy efficiency (EE) of the MTW-MMR systems experiencing both imperfect channel estimation and channel aging based on the four proposed power-scaling schemes. Furthermore, we compare the performance of the full-duplex (FD)-assisted MTW-MMR systems with that of the half-duplex (HD)-assisted MTW-MMR systems, showing that the FD mode outperforms the HD mode with the interference level not exceeding -15 dB. Our studies show that channel aging may significantly degrade the achievable performance in terms of SE and EE. However, the inter-pair interference caused by the other user pairs and the self-interference can be effectively mitigated by the MRC/MRT processing or ZFR/ZFT processing at the relay.