Neural video codecs (NVCs) have seen fast-paced advancement in recent years and already perform close to state-of-the-art traditional video codecs like H.266/VVC. However, NVC investigations have so far focused on improving performance for classical perspective video leaving the increasingly important 360-degree video format unexplored. In this paper, we address this issue and present how existing NVCs can be optimized for 360-degree video while also improving performance on perspective video. As no suitable datasets for neural 360-degree video compression exist, we publish a large-scale 360-degree video dataset consisting of more than 6000 user generated 9-frame sequences with resolutions ranging from 0.5K to 8K. We propose a novel method for training data augmentation exploiting the spherical characteristics of 360-degree video that shows to be crucial for achieving maximum compression performance. An additional positional feature encoding further supports the NVC in dynamic bitrate allocation notably improving the performance for both 360-degree and perspective video. Overall, we achieve rate savings of almost 8% for 360-degree video and more than 3% for perspective video with minimal complexity overhead. The dataset is available at: {link will be provided upon acceptance}. Source code and pre-trained model weights are available at: {link will be provided upon acceptance}.