In the context of Industry 4.0, ensuring the compatibility of digital twins (DTs) with existing software systems in the manufacturing sector presents a significant challenge. The Asset Administration Shell (AAS), conceptualized as the standardized DT for an asset, offers a powerful framework that connects the DT with the established software infrastructure through interoperable knowledge representation. Although the IEC 63278 series specifies the AAS metamodel, it lacks a matching strategy for automating the mapping between proprietary data from existing software and AAS information models. Addressing this gap, we introduce a novel dual data mapping system (DDMS) that utilizes a fine-tuned open-source large language model (LLM) for entity matching. This system facilitates not only the mapping between existing software and AAS models but also between AAS models and standardized vocabulary dictionaries, thereby enhancing the model's semantic interoperability. A case study within the injection molding domain illustrates the practical application of DDMS for the automated creation of AAS instances, seamlessly integrating the manufacturer's existing data. Furthermore, we extensively investigate the potential of fine-tuning decode-only LLMs as generative classifiers and encoding-based classifiers for the entity matching task. To this end, we establish two AAS-specific datasets by collecting and compiling AAS-related resources. In addition, supplementary experiments are performed on general entity-matching benchmark datasets to ensure that our empirical conclusions and insights are generally applicable. The experiment results indicate that the fine-tuned generative LLM classifier achieves slightly better results, while the encoding-based classifier enables much faster inference. Furthermore, the fine-tuned LLM surpasses all state-of-the-art approaches for entity matching, including GPT-4 enhanced with in-context learning and chain of thoughts. This evidence highlights the effectiveness of the proposed DDMS in bridging the interoperability gap within DT applications, offering a scalable solution for the manufacturing industry.