Few-Shot Named Entity Recognition (FS-NER) can identify entity boundaries and types with a limited set of labeled training instances. Pre-trained and large language models have made significant progress in this competitive task. However, accurately recognizing entity boundaries remains challenging due to the semantic ambiguity between entities and non-entities. Additionally, effectively resolving semantic distinctions between entities across various granularities for precise type classification presents another bottleneck. To address these limitations, we propose MBA-NER, a Multi-granularity Boundary-Aware contrastive enhanced framework for two-stage FS-NER. Specifically, in the entity boundary detection stage, MBA-NER first obtains global span representations encoding cross-boundary information. It then enhances these boundary-aware representations through contrastive learning between positive and negative span samples, helping capture and amplify boundary semantics. In the entity type classification stage, MBA-NER constructs cross-granularity semantic representations of entities and further boosts child-granularity entity recognition via contrastive learning across parent-child level granularities. Then, a five-tuple contrastive loss is adopted to jointly optimize the two subtasks, eliminating erroneous entity boundaries and type noise. Extensive experiments on two public FS-NER benchmarks, Few-NERD and CrossNER, demonstrate the superior performance of MBA-NER over existing baselines, especially in scenarios involving multi-granularity entity classification.