Can we quantify the "degree of quantumness" in a quantum algorithm ?
It depends on the context.

 For quantum algorithms, the situation is tricky, since for all we know, P=BPP=BQP.  So we can never say that a quantum algorithm does something that no classical algorithm can do; only something that a naive simulation would have trouble with.  For example, if a quantum circuit is drawn as a graph, then there is a classical simulation that runs in time exponential in the treewidth of the graph).  So treewidth can be thought of as an upper bound to 'quantumness', although not a precise measure.

Sometimes measuring quantumness in algorithms gets conflated with trying to measure the amount of entanglement produced by an algorithm, but we now think that a noisy quantum computer could have computational advantages over classical computer even with so much noise that its qubits are never in an entangled state (e.g. the one clean qubit model).  So the consensus is now more on the side of thinking of the quantumness in quantum algorithms as related to the dynamics rather than the states generated along the way.  This can help explain why 'dequantizing' is not likely to be generally possible.

 For bipartite quantum states, where the context is two-party correlations, we have many many good measures of quantumness.  Many have flaws, like being NP-hard, or not additive, but nevertheless we have a pretty sophisticated understanding of this situation.  Here is a recent review.

 There are other contexts, such as when we have a quantum state and would like to choose between different incompatible measurements.  In this setting, there are uncertainty principles that tell us things about how incompatible the measurements are.  The more incompatible the measurements are, the more 'quantum' a situation we have.  This is related to cryptography and zero-error capacities of noisy channels, among many other things.