Recent neural news recommenders (NNRs) extend content-based recommendation (1) by aligning additional aspects (e.g., topic, sentiment) between candidate news and user history or (2) by diversifying recommendations w.r.t. these aspects. This customization is achieved by ``hardcoding`` additional constraints into the NNR's architecture and/or training objectives: any change in the desired recommendation behavior thus requires retraining the model with a modified objective. This impedes widespread adoption of multi-aspect news recommenders. In this work, we introduce MANNeR, a modular framework for multi-aspect neural news recommendation that supports on-the-fly customization over individual aspects at inference time. With metric-based learning as its backbone, MANNeR learns aspect-specialized news encoders and then flexibly and linearly combines the resulting aspect-specific similarity scores into different ranking functions, alleviating the need for ranking function-specific retraining of the model. Extensive experimental results show that MANNeR consistently outperforms state-of-the-art NNRs on both standard content-based recommendation and single- and multi-aspect customization. Lastly, we validate that MANNeR's aspect-customization module is robust to language and domain transfer.