Accurate and robust knowledge of battery internal states and parameters is a prerequisite for the safe, efficient, and reliable operation of electric flying cars. Battery states such as state of charge (SOC), state of temperature (SOT), and state of power (SOP) are of particular interest for urban air mobility (UAM) applications. This article proposes a new safety-oriented multi-state estimation framework for collaboratively updating the SOC, SOT, and SOP of lithium-ion batteries under typical UAM mission profiles that explicitly incorporates the underlying interplay among these three states. Specifically, the SOC estimation is performed by combining an adaptive extended Kalman filter with a timely calibrated battery electrical model, and the key temperature information, including the volume-averaged temperature, highest temperature, and maximum temperature difference, is estimated using an adaptive Kalman filter based on a simplified 2-D spatially-resolved thermal model. The SOC and SOT estimations interact with the temperature dependence of electrical parameters and the time-varying heat generation. Afterward, the dynamic power capability estimation is fulfilled considering multiple operational constraints such as current, voltage, SOC, and temperature. Both instantaneous and continuous peak power are covered for practical applications, and a novel temperature-constrained peak power estimation method is incorporated to enable more accurate and reliable SOP estimation. The proposed SOC-SOT-SOP joint estimation framework is validated thoroughly by computer simulation and the results demonstrate satisfactory accuracy and resilience.