This paper presents a probabilistic framework to include the effects of both aleatory and epistemic uncertainty sources in coupled multidisciplinary analysis (MDA). A likelihood-based decoupling approach has been previously developed for probabilistic analysis of multidisciplinary systems, but only with aleatory uncertainty in the inputs. This paper extends this approach to incorporate the effects of epistemic uncertainty arising from data uncertainty and model errors. Data uncertainty regarding input variables (due to sparse and interval data) is included through parametric or nonparametric distributions using the principle of likelihood. Model error is included in MDA through an auxiliary variable approach based on the probability integral transform. In the presence of natural variability, data uncertainty, and model uncertainty, the proposed methodology is employed to estimate the probability density functions (PDFs) of coupling variables as well as the subsystem and system level outputs that satisfy interdisciplinary compatibility. Global sensitivity analysis (GSA), which has previously considered only aleatory inputs and feedforward or monolithic problems, is extended in this paper to quantify the contribution of model uncertainty in feedback-coupled MDA by exploiting the auxiliary variable approach. The proposed methodology is demonstrated using a mathematical MDA problem and an electronic packaging application example featuring coupled thermal and electrical subsystem analyses. The results indicate that the proposed methodology can effectively quantify the uncertainty in MDA while maintaining computational efficiency.