Reducing radiation doses benefits patients, however, the resultant low-dose computed tomography (LDCT) images often suffer from clinically unacceptable noise and artifacts. While deep learning (DL) shows promise in LDCT reconstruction, it requires large-scale data collection from multiple clients, raising privacy concerns. Federated learning (FL) has been introduced to address these privacy concerns; however, current methods are typically tailored to specific scanning protocols, which limits their generalizability and makes them less effective for unseen protocols. To address these issues, we propose SCAN-PhysFed, a novel SCanning- and ANatomy-level personalized Physicis-Driven Federated learning paradigm for LDCT reconstruction. Since the noise distribution in LDCT data is closely tied to scanning protocols and anatomical structures being scanned, we design a dual-level physics-informed way to address these challenges. Specifically, we incorporate physical and anatomical prompts into our physics-informed hypernetworks to capture scanning- and anatomy-specific information, enabling dual-level physics-driven personalization of imaging features. These prompts are derived from the scanning protocol and the radiology report generated by a medical large language model (MLLM), respectively. Subsequently, client-specific decoders project these dual-level personalized imaging features back into the image domain. Besides, to tackle the challenge of unseen data, we introduce a novel protocol vector-quantization strategy (PVQS), which ensures consistent performance across new clients by quantifying the unseen scanning code as one of the codes in the scanning codebook. Extensive experimental results demonstrate the superior performance of SCAN-PhysFed on public datasets.