Articulated objects are ubiquitous and important in robotics, AR/VR, and digital twins. Most self-supervised methods for articulated object modeling reconstruct discrete interaction states and relate them via cross-state geometric consistency, yielding representational fragmentation and drift that hinder smooth control of articulated configurations. We introduce PD$^{2}$GS, a novel framework that learns a shared canonical Gaussian field and models the arbitrary interaction state as its continuous deformation, jointly encoding geometry and kinematics. By associating each interaction state with a latent code and refining part boundaries using generic vision priors, PD$^{2}$GS enables accurate and reliable part-level decoupling while enforcing mutual exclusivity between parts and preserving scene-level coherence. This unified formulation supports part-aware reconstruction, fine-grained continuous control, and accurate kinematic modeling, all without manual supervision. To assess realism and generalization, we release RS-Art, a real-to-sim RGB-D dataset aligned with reverse-engineered 3D models, supporting real-world evaluation. Extensive experiments demonstrate that PD$^{2}$GS surpasses prior methods in geometric and kinematic accuracy, and in consistency under continuous control, both on synthetic and real data.