Fine-grained labor market analysis increasingly relies on mapping unstructured job advertisements to standardized skill taxonomies such as ESCO. This mapping is naturally formulated as an Extreme Multi-Label Classification (XMLC) problem, but supervised solutions are constrained by the scarcity and cost of large-scale, taxonomy-aligned annotations--especially in non-English settings where job-ad language diverges substantially from formal skill definitions. We propose a zero-shot skill extraction framework that eliminates the need for manually labeled job-ad training data. The framework uses a Large Language Model (LLM) to synthesize training instances from ESCO definitions, and introduces hierarchically constrained multi-skill generation based on ESCO Level-2 categories to improve semantic coherence in multi-label contexts. On top of the synthetic corpus, we train a contrastive bi-encoder that aligns job-ad sentences with ESCO skill descriptions in a shared embedding space; the encoder augments a BERT backbone with BiLSTM and attention pooling to better model long, information-dense requirement statements. An upstream RoBERTa-based binary filter removes non-skill sentences to improve end-to-end precision. Experiments show that (i) hierarchy-conditioned generation improves both fluency and discriminability relative to unconstrained pairing, and (ii) the resulting multi-label model transfers effectively to real-world Chinese job advertisements, achieving strong zero-shot retrieval performance (F1@5 = 0.72) and outperforming TF--IDF and standard BERT baselines. Overall, the proposed pipeline provides a scalable, data-efficient pathway for automated skill coding in labor economics and workforce analytics.