Functional ANOVA (FANOVA) is a widely used variance-based sensitivity analysis tool. However, studies on functional-output FANOVA remain relatively scarce, especially for black-box computer experiments, which often involve complex and nonlinear functional-output relationships with unknown data distribution. Conventional approaches often rely on predefined basis functions or parametric structures that lack the flexibility to capture complex nonlinear relationships. Additionally, strong assumptions about the underlying data distributions further limit their ability to achieve a data-driven orthogonal effect decomposition. To address these challenges, this study proposes a functional-output orthogonal additive Gaussian process (FOAGP) to efficiently perform the data-driven orthogonal effect decomposition. By enforcing a conditional orthogonality constraint on the separable prior process, the proposed functional-output orthogonal additive kernel enables data-driven orthogonality without requiring prior distributional assumptions. The FOAGP framework also provides analytical formulations for local Sobol' indices and expected conditional variance sensitivity indices, enabling comprehensive sensitivity analysis by capturing both global and local effect significance. Validation through two simulation studies and a real case study on fuselage shape control confirms the model's effectiveness in orthogonal effect decomposition and variance decomposition, demonstrating its practical value in engineering applications.