Traditional image matching methods rely heavily on gradient or intensity information. However, the severe nonlinear radiometric distortion (NRD) between infrared and visible images hinders the extraction of repeatable feature points, leading to poor matching performance. To address this, we propose Shape2Match, a novel framework that replaces point features with more consistent, modality-invariant shape features. Specifically, the method utilizes EfficientSAM to extract shape contours and employs elliptic fourier descriptors (EFD) to parameterize and normalize them, creating shape descriptor that is invariant to translation, rotation, and scale. Shape2Match adopts a coarse-to-fine hierarchical strategy: it first performs robust global shape matching using a weighted EFD distance, followed by precise keypoint matching—using Shape Context—within the coarsely aligned shape pairs. We validated Shape2Match on 153 image pairs from 6 datasets, comparing it against methods like SIFT, RIFT, and MS-HLMO. Experimental results demonstrate that Shape2Match achieves a 100% success rate (SR) across all datasets and significantly outperforms other methods in the number of correct matches (NCM), proving its effectiveness and robustness against NRD, rotation, and scale variations.