Disaggregated performance metrics across demographic groups are a hallmark of fairness assessments in computer vision. These metrics successfully incentivized performance improvements on person-centric tasks such as face analysis and are used to understand risks of modern models. However, there is a lack of discussion on the vulnerabilities of these measurements for more complex computer vision tasks. In this paper, we consider multi-label image classification and, specifically, object categorization tasks. First, we highlight design choices and trade-offs for measurement that involve more nuance than discussed in prior computer vision literature. These challenges are related to the necessary scale of data, definition of groups for images, choice of metric, and dataset imbalances. Next, through two case studies using modern vision models, we demonstrate that naive implementations of these assessments are brittle. We identify several design choices that look merely like implementation details but significantly impact the conclusions of assessments, both in terms of magnitude and direction (on which group the classifiers work best) of disparities. Based on ablation studies, we propose some recommendations to increase the reliability of these assessments. Finally, through a qualitative analysis we find that concepts with large disparities tend to have varying definitions and representations between groups, with inconsistencies across datasets and annotators. While this result suggests avenues for mitigation through more consistent data collection, it also highlights that ambiguous label definitions remain a challenge when performing model assessments. Vision models are expanding and becoming more ubiquitous; it is even more important that our disparity assessments accurately reflect the true performance of models.