I think the undeniable answer is that the Aharonov-Bohm (A-B) effect is a non-QFT effect. By this I mean that QFT doesn't really add anything new to A-B unlike, say, to issues around gauge symmetries (anomalies) or renormalization effects such as the Lamb shift.
However, whether the effect is "classical" or "quantum" is really a question of semantics (as already remarked by Nanite). Most people would say that the particle nature of an electron is "classical" and that interference such as in the double slit experiment is "quantum". On the other hand, interference of light as computed from Maxwell's equations would be called by most as "classical" and particle-like behaviour of light such as the photoelectric effect would then be "quantum".
All and all it seems that the most accurate characterization of "quantum" and "classical" is actually "as understandable by the tradition of 19th century physics" and "as not understandable by 19th century physics". (Understand-able meaning that special and general relativity still somehow falls into the "classical" realm.)
But if we take a look at A-B we see that by this key it is definitely a quantum effect. That is simply because 19th century (the same as us today) knew only massive charged particles for which interference effects can be definitely classified as quantum. And as A-B presents a correction to interference, it cannot be in any way classical.
But yes, I see where you are going by your question on classical QED solutions so I am going to give a more fundamental argumentation why A-B is actually quantum even in this light.
Start by formulating a classical field theory of a field $\phi$. Now you want this field to be somehow charged. How do you do this? You will obviously try to model the interaction by using $\vec{B}$ and $\vec{E}$ because these are the physical components of the electromagnetic field. You do not get A-B.
OK, you say, what about local gauge theory, that gives me minimal coupling to $A^\mu$, does it not? But for that your field must be complex or alternatively be composed from two very particularly coupled real fields. The physics of the complex/two-part field is that there is an antiparticle to your particle. How would you support your argument for a complex field in classical theory?
In summary, without the "quantum" you would most probably not even formulate the right field theory in which the A-B effect happens. But moving forward, another question in my opinion very poorly addressed in standard QFT courses is why would you even believe that a QED Lagrangian represents any physics of particles at all. There are basically two approaches to this question, the "second quantization" narrative (for which, in my humble opinion, people should be condemned to infinite suffering), or the many-particle construction (for which people should receive infinite praise).
The second quantization narrative basically says: we took the Schrödinger equation and made it relativistic but it has problems, so we try to quantize the relativistic quantum mechanics as as a field theory and by chance it turns out okay and - Abrakadabra! - we give the quantization a many-particle interpretation. The field-theoretic Lagrangian is a mere ansatz and the meaning of the classical solutions and their relation to one-particle behaviour is undetermined before quantization. Even though you get an analogy of the A-B effect, you have no way to connect this to one-electron behaviour unless you use the conclusions obtained after quantization. But is it then really a classical effect if you use "after-quantization" development of the theory? Alternatively, you close your eyes on negative energy and all and interpret the QED Lagrangian as one-particle relativistic quantum mechanics but then, obviously, the effect should be considered as quantum.
On the other hand, many-particle construction of QFT says: we know that particles behave in a quantum way but they also interact, get created and destroyed, and also, special relativity. By investigating representation theory, many-particle quantum mechanics and by requiring causality you find out that the interactions and evolution of particles are most practically written down via a particular sum of creation and annihilation operators usually called the field operator and it's formal derivatives. The field operator is, "by chance", identical to the quantized field of relativistic quantum mechanics and the "classical solutions" are only a particular interesting structure which, however, is by no means "classical".
In conclusion, even from a very fundamental QFT-point of view, A-B effect always turns out to be at least a little bit "quantum".
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