The integration of immersive communication into a human-centric ecosystem has intensified the demand for sophisticated Human Digital Twins (HDTs) driven by multifaceted human data. However, the effective construction of HDTs faces significant challenges due to the heterogeneity of data collection devices, the high energy demands associated with processing intricate data, and concerns over the privacy of sensitive information. This work introduces a novel biologically-inspired (bio-inspired) HDT framework that leverages Brain-Computer Interface (BCI) sensor technology to capture brain signals as the data source for constructing HDT. By collecting and analyzing these signals, the framework not only minimizes device heterogeneity and enhances data collection efficiency, but also provides richer and more nuanced physiological and psychological data for constructing personalized HDTs. To this end, we further propose a bio-inspired neuromorphic computing learning model based on the Spiking Neural Network (SNN). This model utilizes discrete neural spikes to emulate the way of human brain processes information, thereby enhancing the system's ability to process data effectively while reducing energy consumption. Additionally, we integrate a Federated Learning (FL) strategy within the model to strengthen data privacy. We then conduct a case study to demonstrate the performance of our proposed twofold bio-inspired scheme. Finally, we present several challenges and promising directions for future research of HDTs driven by bio-inspired technologies.