Online map generation and trajectory prediction are critical components of the autonomous driving perception-prediction-planning pipeline. While modern vectorized mapping models achieve high geometric accuracy, they typically treat map estimation as a deterministic task, discarding structural uncertainty. Existing probabilistic approaches often rely on diagonal covariance matrices, which assume independence between points and fail to capture the strong spatial correlations inherent in road geometry. To address this, we propose a structured probabilistic formulation for online map generation. Our method explicitly models intra-element dependencies by predicting a dense covariance matrix, parameterized via a Low-Rank plus Diagonal (LRPD) covariance decomposition. This formulation represents uncertainty as a combination of a low-rank component, which captures global spatial structure, and a diagonal component representing independent local noise, thereby capturing geometric correlations without the prohibitive computational cost of full covariance matrices. Evaluations on the nuScenes dataset demonstrate that our uncertainty-aware framework yields consistent improvements in online map generation quality compared to deterministic baselines. Furthermore, our approach establishes new state-of-the-art performance for map-based motion prediction, highlighting the critical role of uncertainty in planning tasks. Code is published under link-available-soon.