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Joint user identification and channel estimation (JUICE) is a main challenge in grant-free massive machine-type communications (mMTC). The sparse pattern in users' activity allows to solve the JUICE as a compressed sensing problem in a multiple measurement vector (MMV) setup. This paper addresses the JUICE under the practical spatially correlated fading channel. We formulate the JUICE as an iterative reweighted $\ell_{2,1}$-norm optimization. We develop a computationally efficient alternating direction method of multipliers (ADMM) approach to solve it. In particular, by leveraging the second-order statistics of the channels, we reformulate the JUICE problem to exploit the covariance information and we derive its ADMM-based solution. The simulation results highlight the significant improvements brought by the proposed approach in terms of channel estimation and activity detection performances.