Category-level pose estimation is a challenging task with many potential applications in computer vision and robotics. Recently, deep-learning-based approaches have made great progress, but are typically hindered by the need for large datasets of either pose-labelled real images or carefully tuned photorealistic simulators. This can be avoided by using only geometry inputs such as depth images to reduce the domain-gap but these approaches suffer from a lack of semantic information, which can be vital in the pose estimation problem. To resolve this conflict, we propose to utilize both geometric and semantic features obtained from a pre-trained foundation model. Our approach projects 2D semantic features into object models as 3D semantic point clouds. Based on the novel 3D representation, we further propose a self-supervision pipeline, and match the fused semantic point clouds against their synthetic rendered partial observations from synthetic object models. The learned knowledge from synthetic data generalizes to observations of unseen objects in the real scenes, without any fine-tuning. We demonstrate this with a rich evaluation on the NOCS, Wild6D and SUN RGB-D benchmarks, showing superior performance over geometric-only and semantic-only baselines with significantly fewer training objects.