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We introduce the bilinear generalized vector approximate message passing (BiG-VAMP) algorithm which jointly recovers two matrices U and V from their noisy product through a probabilistic observation model. BiG-VAMP provides computationally efficient approximate implementations of both max-sum and sumproduct loopy belief propagation (BP). We show how the proposed BiG-VAMP algorithm recovers different types of structured matrices and overcomes the fundamental limitations of other state-of-the-art approaches to the bilinear recovery problem, such as BiG-AMP, BAd-VAMP and LowRAMP. In essence, BiG-VAMP applies to a broader class of practical applications which involve a general form of structured matrices. For the sake of theoretical performance prediction, we also conduct a state evolution (SE) analysis of the proposed algorithm and show its consistency with the asymptotic empirical mean-squared error (MSE). Numerical results on various applications such as matrix factorization, dictionary learning, and matrix completion demonstrate unambiguously the effectiveness of the proposed BiG-VAMP algorithm and its superiority over stateof-the-art algorithms. Using the developed SE framework, we also examine (as one example) the phase transition diagrams of the matrix completion problem, thereby unveiling a low detectability region corresponding to the low signal-to-noise ratio (SNR) regime.