Structural health monitoring of floating offshore wind Turbines (FOWTs) is critical for ensuring operational safety and efficiency. However, identifying damage in components like mooring systems from limited sensor data poses a challenging inverse problem, often characterized by multiple solutions where various damage states could explain the observed response. To overcome this, we propose a variational autoencoder (VAE) architecture, where the encoder approximates the inverse operator that maps the observed response to the system's condition, while the decoder approximates the forward operator that maps the system's condition and measured excitation to its response. Conventional Gaussian mixture models can be restrictive and are often computationally prohibitive when used within VAE. This work addresses these limitations by proposing a novel copula-based VAE architecture that decouples the marginal distribution of variables from their dependence structure, providing a flexible method for representing complex, correlated posterior distributions. In our method, the observed response corresponds to statistical features derived from short-term rotation motion signals of the FOWT platform. The damage condition is described by the severity level of two damage classes frequently found in the mooring system (anchoring and biofouling). We provide a comprehensive comparison of the copula against standard Gaussian and Gaussian mixture approaches, considering both diagonal and full covariance matrices, to demonstrate the benefits of copulas. Our analysis, conducted on a high-fidelity synthetic dataset, demonstrates that the Gaussian Copula VAE offers a promising and tractable solution in high-dimensional spaces. Although 2D, the number of copula parameters grows much more slowly with the dimension than the other methods, and therefore it shows promise for higher-D scalability. In the test experiments, the copula achieves superior performance with significantly fewer parameters than the Gaussian Mixture alternatives, whose parametrization grows prohibitively with the dimensionality of the latent space. The results highlight the potential of copula-based VAEs as a powerful tool for uncertainty-aware damage identification in FOWT mooring systems.