Fluid interactions permeate daily human activities, with properties like density and viscosity playing pivotal roles in household tasks. While density estimation is straightforward through Archimedes' principle, viscosity poses a more intricate challenge, especially given the varied behaviors of Newtonian and non-Newtonian fluids. These fluids, which differ in their stress-strain relationships, are delineated by specific constitutive models such as the Carreau, Cross, and Herschel-Bulkley models, each possessing unique viscosity parameters. This study introduces a novel differentiable fitting framework, DiffStir, tailored to identify key physics parameters via the common daily operation of stirring. By employing a robotic arm for stirring and harnessing a differentiable Material Point Method (diffMPM)-based simulator, the framework can determine fluid parameters by matching observations from both the simulator and the real world. Recognizing the distinct preferences of the aforementioned constitutive models for specific fluids, an online strategy was adopted to adaptively select the most fitting model based on real-world data. Additionally, we propose a refining neural network to bridge the sim-to-real gap and mitigate sensor noise-induced inaccuracies. Comprehensive experiments were conducted to validate the efficacy of DiffStir, showcasing its precision in parameter estimation when benchmarked against reported literature values. More experiments and videos can be found in the supplementary materials and on the website: https://sites.google.com/view/diffstir.