The creation and use of Digital Twins of existing structures, such as bridges, implies precise digital replicas that accurately mirror their physical counterparts. Ensuring the trustworthiness of Digital Twins and facilitating informed decision-making necessitates a robust approach to Uncertainty Quantification (UQ). A suitable model-updating scheme is key in preserving the quality and robustness of simulation-based Digital Twins. Model bias, stemming from discrepancies between computational models and real-world systems, poses a significant challenge in achieving this goal. This study delves into the challenges posed by model bias within Bayesian updating of Digital Twins of bridges. Two alternative model bias identification methods —a modularized version of Kennedy and O'Hagan's approach [1] and another one based on Orthogonal Gaussian Processes [2]— are evaluated in comparison with the classical Bayesian inference framework. A key innovation lies in the modification of the aforementioned approaches to incorporate additional information into the Digital Twin framework via the bias term. This enables the extension of the model non-intrusively, leveraging large pools of data inherent in Digital Twins. The study showcases the potential of this approach to correct predictions, quantify uncertainties, and enhance the system with previously untapped information. This underscores the importance of leveraging available data within Digital Twins to identify deficiencies and guide potential future model improvements.