With the rise of collaborative robots, the need for safe, reliable, and efficient physical human-robot interaction (pHRI) has grown. High-performance pHRI requires robust and stable controllers suitable for multiple degrees of freedom (DoF) and highly nonlinear robots. In this article, we describe a cascade-loop pHRI controller, which relies on human force and pose measurements and can adapt to varying robot dynamics online. It can also adapt to different users and simplifies the interaction by making the robot behave according to a prescribed dynamic model. In our controller formulation, two neural networks (NNs) in the "outer-loop" predict human motion intent and estimate a reference trajectory for the robot that the "inner-loop" controller follows. The inner-loop imposes a prescribed error dynamics (PED) with the help of a model-free neuroadaptive controller (NAC), which uses a NN to feedback linearize the robot dynamics. Lyapunov stability analysis gives weight tuning laws that guarantee that the error signals are bounded and the desired reference trajectory is achieved. Our control scheme was implemented on a Personal Robot 2 robot and validated through an exploratory experimental study in point-to-point collaborative motion. Results indicate fast convergence of our controller, and the resulting tracking error, motion jerk, and human control effort are comparable with other methods that require prior training, knowledge, and calibration.