The rapid progress of text-to-image diffusion models raises significant concerns regarding the unauthorized reproduction of trademarked content. While prior work targets general concepts (e.g., styles, celebrities), it fails to address specific brand identifiers. Brand recognition is multi-dimensional, extending beyond explicit logos to encompass distinctive structural features (e.g., a car's front grille). To tackle this, we introduce unbranding, a novel task for the fine-grained removal of both trademarks and subtle structural brand features, while preserving semantic coherence. We construct a benchmark dataset and introduce a novel evaluation framework combining Vision Language Models (VLMs) with segmentation-based classifiers trained on human annotations of logos and trade dress features, addressing the limitations of existing brand detectors that fail to capture abstract trade dress. Furthermore, we observe that newer, higher-fidelity systems (SDXL, FLUX) synthesize brand identifiers more readily than older models, highlighting the urgency of this challenge. Our results confirm that unbranding is a distinct problem requiring specialized techniques. Project Page: https://gmum.github.io/UNBRANDING/.