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Here we show how Feynman's simplified model for the Josephson junction, as a macroscopic two-state coupled quantum system, has a one-to-one correspondence with the stable dynamics of two interfacial Rossby waves in piecewise linear shear flows. The conservation of electric charge and energy of the superconducting electron gas layers become respectively equivalent to the conservation of wave action and pseudoenergy of the Rossby waves. Quantum-like tunneling is enabled via action-at-a-distance between the two Rossby waves. Furthermore, the quantum-like phenomena of avoided crossing between eigenstates, described by the Klein-Gordon equation, is obtained as well in the classical shear flow system. In the latter, it results from the inherent difference in pseudoenergy between the in-phase and anti-phased normal modes of the interfacial waves. This provides an intuitive physical meaning to the role of the wavefunction's phase in the quantum system. A partial analog to the quantum collapse of the wavefunction is also obtained due to the existence of a separatrix between "normal mode regions of influence" on the phase plane, describing the system's dynamics. As for two-state quantum bits (qubits), the two-Rossby wave system solutions can be represented on a Bloch sphere, where the Hadamard gate transforms the two normal modes/eigenstates into an intuitive computational basis in which only one interface is occupied by a Rossby wave. Yet, it is a classical system which lacks exact analogs to collapse and entanglement, thus cannot be used for quantum computation, even in principle.