Uncertainty Quantification (UQ) is paramount for inference in engineering. A common inference task is to recover full-field information of physical systems from a small number of noisy observations, a usually highly ill-posed problem. Sharing information from multiple distinct yet related physical systems can alleviate this ill-posedness. Critically, engineering systems often have complicated variable geometries prohibiting the use of standard multi-system Bayesian UQ. In this work, we introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion. Following a ''learn first, observe later'' paradigm, GABI distills information from large datasets of systems with varying geometries, without requiring knowledge of governing PDEs, boundary conditions, or observation processes, into a rich latent prior. At inference time, this prior is seamlessly combined with the likelihood of a specific observation process, yielding a geometry-adapted posterior distribution. Our proposed framework is architecture-agnostic. A creative use of Approximate Bayesian Computation (ABC) sampling yields an efficient implementation that utilizes modern GPU hardware. We test our method on: steady-state heat over rectangular domains; Reynolds-Averaged Navier-Stokes (RANS) flow around airfoils; Helmholtz resonance and source localization on 3D car bodies; RANS airflow over terrain. We find: the predictive accuracy to be comparable to deterministic supervised learning approaches in the restricted setting where supervised learning is applicable; UQ to be well calibrated and robust on challenging problems with complex geometries.