I gather you are asking about similarities and differences between hadronic and neutrino neutral particle oscillation. There are so many questions packed within your units, that I'll try to answer isolated facets of the jumble, apologizing, with Pascal (Je n’ai fait celle-ci plus longue que parce que je n’ai pas eu le loisir de la faire plus courte), that I lacked the time to make the answers shorter; and leave it to you to fit the facets together.
For starters, both oscillation phenomena are due to quantum-mechanical interference of complex propagation amplitudes, and are only possible due to QM. In both cases, the propagating Hamiltonian (mass) eigenstates ($K_L,~ K_S$, $B_L, ~B_S$; $\nu_{1,2,3}$) are not the interaction combinations which production mechanisms and detectors rely on, so the mismatches in time and hence distance of the ever-shifting interaction combination states are described as an oscillation.
(There are theological discussions about preferences in momentum vs energy eigenstates involved in the parallel propagation, but just don't go there.)
The differences between the two cases are just in the microscopic QM description of these amps. You wrote down the diagrams for Bottomness oscillations of mesons.
Why are these processes called flavor-changing neutral currents? It is because there is no electric charge transfer? So what are examples of flavour changing charged currents?
The standard plinth of the QFT edifice is the current, in this context a bilinear of quarks coupling to mesons, for example, and to something bosonic, like a Z or a W, or a grouping $W^+W^-$. The net charge of the object makes the difference between a charged current, versus a neutral current. In your case, the neutral currents involved are
$\bar d \Gamma b_L$ and $\bar b \Gamma d_L$ , where I've left the tensor structure of Γ vague/open, and you may construct them by pinching charged currents, as in your loop diagrams. They are not flavor neutral, so the amp you wrote changes bottomness by two. Because of the CKM/PMNS construction, charged currents can and do change flavor, on account of the built-in intergeneration straddling of the W couplings.
Are processes involving the $Z$ boson allowed? (I expect the answer is no because the weak neutral interaction couples fermions of the same type)
Right you are. If the neutral current couplings to the Z were not diagonal in the mass eigenstates, the unitary basis change to those would cancel itself in the couplings. But now you have exhausted this freedom, and there is nothing left to make the above bottomness-violating term flavor diagonal.
In the lepton sector, there is the $U_{PMNS}$ matrix. So, is it also possible to have diagrams where "oscillation" with flavour violation can occur (e.g. we have internal lepton line and external lines with neutrinos)?
Yes, this is the point behind neutrino oscillations, except you got the internal particle wrong (the "oscillating"/propagating particle has to be neutral, so the neutrinos, as above), and you don't need a loop: it is a tree diagram. Here is a trail map on how to convert the snipped upper part of you second diagram to a neutrino production+oscillation+detection one.
A $\pi^+$ decays at the production vertex to a virtual $W^+$ which converts to $\nu_\mu$ and $\mu^+$, lost at production. But $\nu_\mu$ is a fiction: it is a superposition of the propagating $\nu_1,\nu_2,\nu_3$, zipping along with infinitesimally small relative delays. At the detection vertex, they suck up a virtual $W^-$ off the detector's nuclei, to turn into a negative lepton, but, this time, on account of the mismatch of velocities in the wave packet, a $e^-$, for the sake of argument. So, this presents as an oscillation ( ⇄ leptonic flavor change) of the bogus $\nu_\mu \to \nu_e$. You are not studying mesons, so you don't need two fermion lines as in B oscillations, one fermion line propagation will do.
Why is neutrino oscillation instead described quantum mechanically?
Explained above: it is all QM.
Is it because we need not suppress the amplitude (unlike the GIM mechanism)?
Actually yes, in a way, if by this you mean "what makes the respective oscillations rare/slow enough to be detectable at macroscopic scales?" Indeed, (do the estimate calculation!), in the hadronic case, the doubly week, GIM suppressed, transition amp is sufficiently slow to give you a macroscopic (lab) effect for the high masses involved; while, for neutrinos, the tree amp suffices, on account of the smallness of neutrino masses: The relevant parameter for the distance of oscillation is L ~ E/m², long baseline.
Best Answer
The 3 neutrino families ($e$, $\mu$, $\tau$) are usually called neutrino flavors.
Neutrino flavor oscillation requires that the mass eigenstates of neutrinos are not equal and that the mass eigenstate is also not a flavor eigenstate. Since a neutrino is always produced in a flavor eigenstate (i.e. associated with an $e$, $\mu$, $\tau$), this flavor eigenstate wave function will be a mixture of the 3 mass eigenstates such that at the time of production it is a pure flavor eigenstate. However as the neutrino wave function propagates, the 3 mass eigenstates will effectively move at different speeds so that at the point in space where the propagating neutrino interacts with the measuring apparatus, it will be a different mixture of flavor eigenstates. Thus the possibility of flavor oscillation requires that the masses of the mass eigenstates are not equal.
That is why the flavor oscillation experiments always measure differences in masses (squared) but not absolute masses. So since oscillations among all three flavors have been observed, there must be 3 different mass eigenstates with different masses. However, the flavor oscillation experiments would allow the lightest mass eigenstate to have zero mass but would require that at least 2 mass eigenstates have masses that are non-zero and are not equal.
It is commonly assumed that all 3 masses are non-zero and not equal.
Note, for example, that you cannot talk about the mass of the electron neutrino since the flavor eigenstate of an electron neutrino will be a mixture of the 3 different mass eigenstates.