The proper constructions resembling quantum mechanic's formalism does exist in classical mechanics, but it goes a bit beyond lagrangian formalism. In classical mechanics, you can represent a system by a phase space with points corresponding to states of the system. Now, functions over that phase space form a symplectic Lie algebra together with the Poisson bracket as the Lie bracket. The Hamilton's equations can than be seen as being the action of the Hamiltonian acting on the phase space function $q_i$ and $p_i$ by the Lie bracket action. What these equations say is that this action of the Hamiltonian is related to the derivative with respect to a parameter (time) of these functions over phase space.
If you place yourself in the Heisenberg point of view of quantum mechanics, where the time-dependance is placed on the operators, we see a direct connection. The Hamilton's equation together with the the Poisson bracket, can be related to the present case replacing the functions over phase space with operators acting on a Hilbert space and replacing the Poisson bracket with the commutator as a Lie bracket. What you get are the equations of motions of quantum mechanics in the Heisenberg picture (if the operators don't depend explicitly on time). It is rather interesting that the exact same algebraic structure of motion (Poisson algebra) is present in classical mechanics and in quantum mechanics (and even QFT! See canonical quantization.), the difference being in the realizations using different mathematical objects.
Now, we know we can rewrite these equations of time dependence using the state vectors in the Hilbert space instead of the operators themselves. This is the Schrödinger picture and leads to the equation you gave. There is also an analogy here with classical mechanics here when you consider that the Hamiltonian can be used to define an Hamiltonian flow on the phase space which is essentially curves on the phase space parametrized by what we call time.
The case with the time-independent Shrödinger equation is a bit different. When you have the time-dependent equation and you want to solve it, you suppose the solutions are separable in terms of the time dependence and the space dependence. And from PDE theory, solving the equation is reduced to solving the time independent equation by separation of variables.