Can classical theories exhibit quantum-like effects? For quite a while in the first half of the 20th century, many physicists tried to concoct some manner of unified theory to explain all known phenomenon, a lot of them using geometric theories (ie teleparallelism, affine gravity, asymmetric metric theories, Kaluza-Klein, etc). While their main goal was the unification of general relativity and electromagnetism, there was also the hope that it would explain quantum phenomenon, by outputting some discrete sets of solutions.
As far as I know, none of them succeeded in that domain. What I want to know is, was there ever any hope of it? Can a classical theory, be it a field theory, of extended objects or geometric, have a discrete spectrum, or does it by its very structure (perhaps via some theorems on differential equations or whatever else) force it to have a continuous spectrum for all variables? Also just in case, did any such theory ever succeed at least in that aspect, even if it failed in other aspects to describe quantum mechanics?
 A: At present they are called deterministic or hidden variable  theories, and they try to show that the standard model elementary particles are composite and there is a deterministic theory from which the standard model and quantum mechanics emerge, similar to the emergence of thermodynamics to statistical mechanics. 
As far as I know none have been successful over the total experimentally supported points of the standard model.
One of the proposals  is by   G. 't Hooft (who is still a member on this site, but inactive for some years)

t Hooft has "deviating views on the physical interpretation of quantum theory".He believes that there could be a deterministic explanation underlying quantum mechanics. Using a speculative model he has argued that such a theory could avoid the usual Bell inequality arguments that would disallow such a local hidden variable theory. In 2016 he published a book length exposition of his ideas

An early theory is Bohmian mechanics, again trying to get the mathematical results of quantum mechanics with an underlying deterministic model.
As far as I know it is successful at non relativistic quantum mechanics, that is why it is called an interpretation of quantum mechanics. Again AFAIK people are still working on trying to find a relativistic version that will do the same.
As for your title:

Can classical theories exhibit quantum-like effects?

The notes coming out of musical instruments are quantum like effects, so yes there are classical equations that give specific energies etc. But to describe quantum theories without the quantum mechanical postulates, by assuming an underlying deterministic  level, is a more difficult story. 
A: You may be interested in Stochastic Electrodynamics :
https://en.wikipedia.org/wiki/Stochastic_electrodynamics
https://arxiv.org/abs/1205.0916
https://arxiv.org/abs/1903.00996
I've read several papers on that classical theory, and I must admit that it shaked strongly my belief in Quantum Mechanics!  Since then, I no longer see QM the same way as before and feel more perplexed about it.
