This article introduces a modular robust subsystem-based adaptive (RSBA) control design with a new stability analysis for a class of uncertain interconnected systems with unknown modeling errors and interactions. First, we propose a nonlinear state observer for this class of systems that ensures uniformly exponential convergence of the estimation error by utilizing a new adaptive term, extending the conventional continuous Luenberger concept. Second, we introduce a novel adaptive subsystem-based control strategy for trajectory tracking, which incorporates an interesting term called the "stability connector", designed to capture dynamic interactions among subsystems during stability analysis, preventing excessive complexity as the system order increases. This represents the first instance of this term allowing modular control with exponential and uniform convergence rates, effectively handling unknown non-triangular nonlinearities. In addition to rigorous theoretical proofs by the Lyapunov theory, two complex systems are explored in simulations to confirm the merits of the suggested control schemes.