On the heels of orthogonal time frequency space (OTFS) modulation, the recently discovered affine frequency division multiplexing (AFDM) is a promising waveform for the sixth-generation wireless network. In this paper, we study the widely-used embedded pilot-aided (EPA) channel estimation in multiple-input multiple-output AFDM (MIMO-AFDM) system with fractional Doppler shifts. We first formulate the vectorized input-output relationship of MIMO-AFDM, and theoretically prove that MIMO-AFDM can achieve full diversity in doubly selective channels. Then we illustrate the implementation of EPA channel estimation in MIMO-AFDM and unveil that serious inter-Doppler interference (IDoI) occurs if we try to estimate the channel gain, delay shift, and Doppler shift of each propagation path. To address this issue, the diagonal reconstructability of AFDM subchannel matrix is studied and a low-complexity embedded pilot-aided diagonal reconstruction (EPA-DR) channel estimation scheme is proposed. The EPA-DR scheme calculates the AFDM effective channel matrix directly without estimating the three channel parameters, eliminating the severe IDoI inherently. Since the effective channel matrix is necessary for MIMO-AFDM receive processing, we believe this is an important step to bring AFDM towards practical communication systems. Finally, we investigate the orthogonal resource allocation of affine frequency division multiple access (AFDMA) system. Simulation results validate the effectiveness of the proposed EPA-DR scheme.