Massive machine-type communication (mMTC) is characterized by the sporadic device activities and angular-sparse channels, creating an opportunity for the efficient joint device activity detection (AD) and channel estimation (CE). However, it is computationally intractable to solve the related 2D-sparse recovery problem. Existing methods often resort to transforming this problem into an 1D-sparse recovery issue, which can result in performance degradation due to energy leakage from the off-grid effect. Moreover, these methods overlook the potential benefits of common sparsity across different frequency bands. To address these limitations, a novel sparse Bayesian learning (SBL) framework for the joint AD and CE is proposed in this article, and two advanced sparsity structures under the frequency multiplex (corresponding to frequency sparsity and partial common sparsity) are additionally exploited to elevate the sparse recovery performance substantially. The key to the success of the proposed method lies in two crucial factors: 1) the employment of an innovative independent variational Bayesian inference (VBI) factorization technique, effectively decoupling the challenging 2D-sparse recovery problem, and mitigating the off-grid mismatch and 2) the introduction of a hybrid sparsity prior into the SBL framework, seamlessly integrating additional frequency sparsity and common sparsity across different active frequency bands. Simulation results verify the superiority of the proposed method and indicate that the proposed method can achieve almost 50% normalized mean-square error performance enhancement compared with the state-of-the-art methods, attributed to its flexible employment of the sophisticated sparsity structure.