We consider channel estimation within pulse-shaping multicarrier multiple-input multiple-output (MIMO) systems transmitting over doubly selective MIMO channels. This setup includes MIMO orthogonal frequency-division multiplexing (MIMO-OFDM) systems as a special case. We show that the component channels tend to exhibit an approximate joint group sparsity structure in the delay-Doppler domain. We then develop a compressive channel estimator that exploits this structure for improved performance. The proposed channel estimator uses the methodology of multichannel group sparse compressed sensing, which combines the methodologies of group sparse compressed sensing and multichannel compressed sensing. We derive an upper bound on the channel estimation error and analyze the estimator's computational complexity. The performance of the estimator is further improved by introducing a basis expansion yielding enhanced joint group sparsity, along with a basis optimization algorithm that is able to utilize prior statistical information if available. Simulations using a geometry-based channel simulator demonstrate the performance gains due to leveraging the joint group sparsity and optimizing the basis.