Accurate alignment of multi-stain liver biopsy whole-slide images is critical for emerging computational pathology workflows, yet remains challenging due to curved biopsy geometry, non-rigid distortions, and substantial stain-dependent appearance variation. We introduce CurvReg, a curvature-aware, stain-independent registration framework designed to address these challenges by decomposing each biopsy into anatomically coherent segments for piecewise affine alignment, followed by deep learning-based deformable refinement operating directly on tissue segmentation masks rather than raw stained images. By estimating deformation from structural geometry rather than stain intensities, CurvReg enforces anatomically plausible warps and substantially improves robustness to staining, imaging, and device variability. We evaluated CurvReg across three multi-stain liver biopsy datasets and an independent public kidney biopsy dataset from the ANHIR challenge. On the internal multi-stain dataset (Dataset #1) containing hematoxylin–eosin (H&E), Masson’s trichrome (MT), and CK7 sections with bile duct annotations, pre-registration mean target registration error (TRE) measured 231.5 μm (H&E–MT), 255.3 μm (MT–CK7), and 188.7 μm (H&E–CK7). After registration, TRE decreased to 51.9 μm, 57.9 μm, and 50.9 μm, respectively, with a >70% reduction across all stain pairs (p < 0.001). Dice similarity increased from 0.54–0.60 to 0.95 (H&E–MT), 0.95 (MT–CK7), and 0.96 (H&E–CK7) (p < 0.001). On the internal portal tract and hepatic vein-annotated H&E–MT dataset (Dataset #2), TRE improved from 147.9 to 48.0 μm (corresponding to a 67.5% reduction; p < 0.001) and Dice from 0.76 to 0.98 (p < 0.001). External validation on a heterogeneous multi-center H&E–MT dataset showed similar gains, with TRE reduced from 93.4 to 42.2 μm (>50% reduction; p < 0.001) and Dice increasing from 0.61 to 0.95 (p < 0.001), confirming robust generalization across scanners, staining protocols, and tissue processing variations. Evaluation on the independent ANHIR kidney biopsy dataset (PAS–MT sections) further demonstrated the generalizability of CurvReg to a different tissue type and stain combination, reducing TRE from 107.2 to 81.6 μm (p = 0.024) while improving Dice similarity from 0.90 to 0.97 (p < 0.001). To highlight downstream clinical value, we demonstrate that MT-derived portal tract and hepatic vein segmentations can be accurately transferred onto H&E using CurvReg, enabling consistent vascular delineation within the diagnostic stain.