The analogy I've worked out from pieces here and there goes like this:
using the logarithm and exponential, we define for two real numbers x and y the following binary operation $x §_h y := h .ln( e^{x/h} + e^{y/h} )$ which depends on some positive real parameter $h$. Then we observe that as $h \rightarrow 0$ the number $x §_h y$ tends to $max(x,y)$. (Proof: assume without loss of generality that $x>y$, so $(y-x)/h <0$. But since $h. ln( e^{x/h} + e^{y/h} ) = h. ln( e^{x/h} . (1+e^{(y-x)/h} )$ as $h \rightarrow 0$ we tend to $h. ln (e^{x/h} . (1+0) ) = x = max(x,y)$. QED.)
Now, in quantum mechanics the canonical commutation relations between positions and momenta operators read $[x_u,p_v] = i \hbar \delta_{uv}$ and in the limit $\hbar \rightarrow 0$ those commutators thus tend to $0$, which says that we recover classical mechanics where everything commutes. And in quantum mechanics what matters are wavefunctions which are superpositions of things of the form $A.e^{iS/\hbar}$ where $A$ is some amplitude and $S$ some phase (the action of the path).
Going back to $§_h$ we can rewrite $e^{(x §_h y)/h } = e^{x/h}+ e^{y/h}$, and so there is your analogy: the tropical mathematics operation max(,) is some kind of classical limit of the (thereby quantum) operation +.
There are two different views about the semiclassical limit in quantum mechanics, the first is based on a somewhat shaky ground due to the fact that the existence of the Feynman integral is not proved yet. On the other side, Wiener integral, its imaginary time counterpart does exist and one could pretend to work things out from this and then move to the Feynman integral. The other approach relies on substantial mathematical theorems due to Elliott Lieb and Barry Simon in the '70 and is essentially valid for many-body physics. These latter results make the limit $\hbar\rightarrow 0$ and $N\rightarrow\infty$ equivalent while the former is not really a physical limit due to the fact that Planck constant is never zero.
Starting from Feynman path integral, the standard formulation applies to a mechanial problem described from a Lagrangian $L$, normally $L=\frac{\dot x^2}{2}-V(x)$ but one can extend this to more general cases, and then the postulate is that, given a path $x(t)$, this must contribute to the full quantum mechanical amplitude of a particle going from the point $x_a$ to $x_b$ with a term $e^{\frac{i}{\hbar}S}$ being $S=\int_{t_a}^{t_b}dtL(\dot x,x,t)$ the action. All the possible paths contribute and so, the full amplitude will be given by the formal writing
$$
A(x_a,x_b)\sim\int[dx(t)]e^{\frac{i}{\hbar}\int_{t_a}^{t_b}dtL(\dot x,x,t)}.
$$
Be warned that this integral is not proved to exist yet, but the Wiener counterpart, that can be obtained changing $t\rightarrow it$, exists and describes Brownian motion. Now, if you take the formal limit $\hbar\rightarrow 0$ to this integral you will immediately recognize the conditions to apply the stationary phase method to it. This implies that the functional must have an extremum and this can be obtained by pretending that
$$
\delta S=\delta \int_{t_a}^{t_b}dtL(\dot x,x,t)=0
$$
that is, the paths that give the greatest contribution are the classical ones and one recover the classical limit as a variational principle as learned from standard textbooks.
While this is a quite common approach, to extend what really happens to a macroscopic system that we can see to respect all the laws of classical mechanics, we have to turn our attention to the limit of a large number of particles $N\rightarrow\infty$. In this case one has more rigorous results. These are due to Lieb and Simon as already said above. They published two papers about
Lieb E. H. and Simon B. 1973 Phys. Rev. Lett. 31, 681.
Lieb E. H. and Simon B. 1977 Adv. in Math. 23, 22.
In the first paper, their theorem 4 states
Theorem: For $\lambda < Z$, let $E_N^0$ and $\rho_N^0(x)$ denote the ground-state energy and one-electron distribution function for N spin-$\frac{1}{2}$ electrons obeying the Pauli principle and interacting with $k$ nuclei as described above. Then (a) $N^{-\frac{7}{3}}E_N^0\rightarrow E_1$, as $N\rightarrow\infty$; (b) $N^{-2}\rho_N^0(N^{-\frac{1}{3}}x)\rightarrow\rho_1(x)$ as $N\rightarrow\infty$, where convergence in (b) means that for any domain $D\subset R^3$, the expected fraction of electrons in $N^{-\frac{1}{3}}D$ approaches $\int_D\rho_1 (x)d^3x$.
Where $\rho_1(x)$ and $E_1$ refer to the Thomas-Fermi distribution and the corresponding energy. This theorem states that the limit $N\rightarrow\infty$ for a quantum system, under some mild conditions, is the Thomas-Fermi distribution. A system with this distribution is a classical system. The fact that a system with a Thomas-Fermi distribution is a classical one can be seen through the following two references:
W. Thirring(Ed.), The Stability of Matter: From Atoms to Stars - Selecta of E. Lieb, Springer-Verlag (1997).
L. Hörmander, Comm. Pure. Appl. Math. 32, 359 (1979).
The second paper just gives the mathematical support to derive Thomas-Fermi approximation as the leading order of a classical expansion for $\hbar\rightarrow 0$ that I will not present here.
Best Answer
It is perhaps helpful to distinguish between four types of mechanics here:
In both the classical and quantum regimes, a mixed state can be viewed as a convex (or classical) superposition of pure states (with a pure classical state $(q,p)$ identified with the Dirac probability density function $\delta_{(q,p)}$, and a pure quantum state $|\psi \rangle$ identified with a pure density matrix $|\psi \rangle \langle \psi|$). So in principle the pure-state mechanics describes the mixed-state mechanics completely (albeit with the caveat that in the quantum case, in contrast to the classical case, the decomposition of a mixed state as a superposition of pure states is non-unique). However, the correspondence principle is clearest to see at the mixed state level, i.e. to compare 2. with 4. in the semiclassical limit $\hbar \to 0$, rather than comparing 1. with 3.. Indeed, any density matrix $\rho$ has a Wigner transform $\tilde \rho$, which is a function on phase space defined via duality as $\int \tilde \rho(q,p) A(q,p)\ dq dp = \hbox{tr}( \rho \hbox{Op}(A) )$ for any classical observable $A$, where $\hbox{Op}(A)$ is the (Weyl) quantisation of $A$ (i.e. the Wigner transform is the adjoint of the quantisation operator). This Wigner transform $\tilde \rho$ will usually not be non-negative, and hence will not be a classical probability density function, but in semiclassical regimes it is often the case that $\tilde \rho$ will tend (in a suitable weak sense) to a classical probability density when $\hbar \to 0$, which will then evolve by the classical advection equation. This is the dual to the assertion that the quantum Heisenberg equation $\partial_t A = \frac{i}{\hbar} [H,A]$ for the evolution of quantum observables converges to the classical Poisson equation $\partial_t A = -\{ H,A\}$ for the evolution of classical observables in the semiclassical limit $\hbar \to 0$.
There is still a correspondence at the level of 1. and 3., but it is a bit trickier to see; one has to restrict to things like "gaussian beam" type solutions $|\psi \rangle$ to the Schrödinger equation that are well localised in both position and momentum space, in order to get a classical limit that is a pure state rather than a mixed state. (An arbitrary wave function would instead get a "phase space portrait" which in the semiclassical limit becomes [assuming some equicontinuity and tightness, and possibly after passing to a subsequence, as noted in comments] a mixed state from 2., rather than a pure state from 1.).