Here is one example, but it's probably remote from the systems you have in mind. Consider a model of spins/moments $S^{\alpha=x,y,z}_j$ living on a lattice with sites $j \in \mathbb{Z}$, subject to a Hamiltonian of form

$H = \sum_{j}[ J_j S^z_j S^z_{j+1}+ \epsilon h_j S^x_j + g_j S^z_j] $

where $J_j,h_j,g_j$ are chosen randomly in $[-1,1]$ and $ \epsilon $ is a small number (but non-zero). Now if you quantize this model, and promote $S^\alpha$ to spin-1/2 operators, one can show that the model **has an infinite set of (quasi)local conserved quantities. **Thus, in a sense it has an infinite set of emergent conservation laws/symmetries.

Some points: 1)There's a proof of the above statement, with one caveat which most people don't worry about (see also this). 2) Quasi-local = the conserved quantities are local, but have very small support on larger operators; the weight of this support on larger operators decays exponentially with the size of the operator. 3) There's a wide class of models which exhibit this sort of behavior; they constitute a phase of matter known as many-body localized (MBL) phases.

Now, in the classical version of the same model, the above construction fails. Indeed, it is believed that the classical version of the model does not possess the local conservation laws I mentioned. This statement is connected to the fact that the KAM theorem for classical systems becomes less tenable as the number of degrees of freedom in a system increases.