Identifying the inertial parameters of real robotic manipulators is a fundamental step towards realistic modeling and better controller performances, which is crucial for safe human-robot interaction. Our work introduces a novel framework for identifying a generalized set of base inertial parameters of a serial link manipulator. This framework is designed to be adaptable to accommodate any new mounting configuration of the robot. Our theoretical analysis highlights the influence of the robot's mounting configuration on the emergence of new parameters that cannot be identified through the conventional vertical base-axis mounting approach studied previously. To validate our proposed framework, we carried out two main experiments: the first involved simulation to establish the feasibility of our concept, and in the second, our framework was employed on a Franka Emika Robot in a real-world scenario to demonstrate and validate our approach. Our simulation results confirmed the feasibility of our proposed framework, while our real-world experiment successfully identified the generalized base inertial parameter set and validated its applicability to a new robot mounting configuration.