Dense retrieval methods have been mostly focused on unstructured text and less attention has been drawn to structured data with various aspects, e.g., products with aspects such as category and brand. Recent work has proposed two approaches to incorporate the aspect information into item representations for effective retrieval by predicting the values associated with the item aspects. Despite their efficacy, they treat the values as isolated classes (e.g., "Smart Homes", "Home, Garden & Tools", and "Beauty & Health") and ignore their fine-grained semantic relation. Furthermore, they either enforce the learning of aspects into the CLS token, which could confuse it from its designated use for representing the entire content semantics, or learn extra aspect embeddings only with the value prediction objective, which could be insufficient especially when there are no annotated values for an item aspect. Aware of these limitations, we propose a MUlti-granulaRity-aware Aspect Learning model (MURAL) for multi-aspect dense retrieval. It leverages aspect information across various granularities to capture both coarse and fine-grained semantic relations between values. Moreover, MURAL incorporates separate aspect embeddings as input to transformer encoders so that the masked language model objective can assist implicit aspect learning even without aspect-value annotations. Extensive experiments on two real-world datasets of products and mini-programs show that MURAL outperforms state-of-the-art baselines significantly.