The human visual system can effortlessly recognize an object under different extrinsic factors such as lighting, object poses, and background, yet current computer vision systems often struggle with these variations. An important step to understanding and improving artificial vision systems is to measure image similarity purely based on intrinsic object properties that define object identity. This problem has been studied in the computer vision literature as re-identification, though mostly restricted to specific object categories such as people and cars. We propose to extend it to general object categories, exploring an image similarity metric based on object intrinsics. To benchmark such measurements, we collect the Common paired objects Under differenT Extrinsics (CUTE) dataset of 18, 000 images of 180 objects under different extrinsic factors such as lighting, poses, and imaging conditions. While existing methods such as LPIPS and CLIP scores do not measure object intrinsics well, we find that combining deep features learned from contrastive self-supervised learning with foreground filtering is a simple yet effective approach to approximating the similarity. We conduct an extensive survey of pre-trained features and foreground extraction methods to arrive at a strong baseline that best measures intrinsic object-centric image similarity among current methods. Finally, we demonstrate that our approach can aid in downstream applications such as acting as an analog for human subjects and improving generalizable re-identification. Please see our project website at https://s-tian.github.io/projects/cute/ for visualizations of the data and demos of our metric.