Which features of quantum physics are not fundamentally quantum but are due to indeterminism?

What is fundamentally quantum? We argue that most of the features, problems, and paradoxes -- such as the measurement problem, the Wigner's friend paradox and its proposed solutions, single particle nonlocality, and no-cloning -- allegedly attributed to quantum physics have a classical analogue if one is to interpret classical physics as fundamentally indeterministic. What really characterizes non-classical effects are incompatible physical quantities, which, in quantum quantum theory are associated to the fundamental constant $\hbar$.