Prefabricated construction is increasingly applied in the construction industry. Recent academic and industrial efforts highlight that Digital Twin as a Service (DTaaS) is an innovative solution to strengthen the advantages of prefabricated construction by enabling continuous monitoring, precise control, and optimized maintenance of each unique module. However, the complexity of digital twins presents significant challenges for both platforms and subcontractors to apply and fully understand this technology. It is, therefore, crucial to develop a well-designed sharing environment where stakeholders volunteer to share their digital twin resources for synergistic benefits. This research proposes an innovative blockchain-based fine-grained digital twin sharing framework, with detailed architecture focused on two key stages of prefabricated construction: on-site construction and logistics involving warehousing and transport. An incentive mechanism is introduced to encourage all the stakeholders engaged in digital twin sharing framework to contribute and share more digital twin resources. Evolutionary and cooperative games explore the effective settings of the incentive policies, aiming to ensure the feasibility and sustainability of the proposed system by motivating users to share digital twins with fine granularity. Mathematical and numerical results demonstrate the optimal subsidy policies with/without budgetary constraints and further highlight the effectiveness of well-crafted incentives. Findings reveal that better performance in terms of profitability and digital twin sharing is achieved under the proposed incentives. Subsidies are primarily allocated to offset costs with a certain budget, whereas the emphasis shifts towards quality improvement under an ample budget. This study lays a solid foundation for digital twin reuse and development.