The Internet of Bio-Nano Things (IoBNT) is a groundbreaking concept in communication and network engineering, aiming to establish connections between biological cells, tissues, nanodevices and the Internet. Within IoBNT, bio-inspired molecular communication (MC) is a promising communication paradigm for facilitating information exchange between biological cells and nanodevices. Blood vessels are essential channels for MC systems in IoBNT within the human body. Typically, cylindrical channels with Newtonian fluid flows are utilized to model vascular channels in MC research. However, blood behaves as a non-Newtonian fluid at low shear rates, resulting in distinct flow velocity profiles that affect the transport of information molecules (IMs). To analyze the impact of non-Newtonian fluid flow velocity profiles on the transmission of IMs and the performance of MC systems within IoBNT, we develop an MC system focusing on a cylindrical channel with non-Newtonian fluid flow. We obtain the distribution of molecules by deriving the cross-sectional average concentration of IMs and formulate the expression for the probability of IMs observed by the receiver. Furthermore, we introduce a simple channel coding scheme to mitigate the influence of intersymbol interference (ISI). Moreover, the impacts of the non-Newtonian fluids on MC systems are analyzed in terms of probabilities of detection and false alarm, probability of error, mutual information, and achievable rate. Finally, simulations are performed to validate the analytical results. The findings in this article highlight the importance of considering the rheological behavior of the fluid in optimizing the system design.