In recent years, the classical Lozi map has attracted considerable attention due to its simple structure and rich dynamical properties. Building upon this foundation, the present work introduces a novel fractional-order memristive extension of the Lozi map by integrating three advanced research directions: discrete-time systems, memristive elements, and fractional calculus. Specifically, the Grunwald–Letnikov fractional difference operator is employed to formulate a new three-dimensional fractional map devoid of equilibrium points. Through numerical analysis, the proposed system reveals the coexistence of multiple hidden chaotic attractors and initial-condition-dependent bifurcation scenarios. A wide range of complex behaviors emerges, demonstrating the system’s high sensitivity to initial states and fractional orders. Furthermore, the dynamical characteristics are investigated using bifurcation diagrams, Lyapunov exponents, and phase portraits. To validate the feasibility of the model, a digital implementation is realized on a microcontroller-based hardware platform, confirming its potential for practical applications in nonlinear signal processing and chaotic systems.