As well known, the representations of the operator algebra of a quantum system fall into superselection sectors, which consist of sub-Hilbert spaces of a quantum state space, on which the physical observables act irreducibly. Each superselection sector may be identified with an elementary system.
There are many types of superselection sectors which are not related to the topology of the configuration space; for example, the mass superselection sectors in Galilean invariant theories, the charge superselection sectors in QED, the superselection sectors of spontaneously broken theories etc.
Inequivalent quantizations, form a special class of superselection sectors, where the quantum theory depends on additional parameters which are either absent or ineffectual in the classical theory, such as the Aharonov-Bohm fluxes.
Precisely; these sectors are in a one to one correspondence with the representations of the fundamental group $\pi_1(Q)$ of the configuration group $Q$. For this reason, they are usually referred to as the topological sectors in contrast to the other superselection sectors which are nontopological. (This is the answer to your second question).
To answer the first question (about the "fictitious" gauge fields), and sketch the proof of the above assertion, let me refer you to this question by AccidentalFourierTransform (the question is about the momentum operator as defined in Bryce DeWitt's book (chapter 11)).
DeWitt describes this type of superselection sectors as follows: When, we quantize, we need to replace classical observables not commuting with the configuration space coordinates by differential operators.
In particular, in order not to change the canonical commutation relations $ [q_i, p_j] = i \delta_{ij}$ the most general form the momentum operator can take is
$$p_i = -i \frac{\partial}{\partial q_i}-\omega_i(q)$$
(In field theory, both $q$ and $p$ are fields and the indices $i$, $j$ can ne continuous, i.e., parametrize some base space).
In order for the momentum-momentum commutation relations $ [p_i, p_j] = 0$not to change, the vector functions $\omega_i(q)$ need to satisfy the relation:
$$\frac{\partial}{\partial q_i}\omega_j(q) - \frac{\partial}{\partial q_j}\omega_i(q) = 0$$
Thus, the vector functions $\omega_i(q)$ must be vector potentials with vanishing field strength, which are called flat connections. Bundles equipped with flat connections are called flat bundles, they are classified by the character representations of the fundamental groups of the configuration space, please see, Kobayashi: Differential geometry of complex vector bundles section 1.2.
Here is a shorthand sketch of the proof. The flat vector potentials (modulo gauge transformations can be characterized by the Wilson loops:
$$e^{i \oint_{\Gamma} \omega}$$
As $\Gamma$ runs over all possible loops in $Q$. However, since, the field strengths are vanishing, the result doesn't depend on the particular loop but on its homotopy class. Since the Wilson loops satisfy the group properties; we obtain that the flat connections are in a one-to-one correspondence with the character, i.e., $U(1)$ representations of the fundamental group $\pi_1(Q)$.
$$\mathrm{Map}(\pi_1(Q), U(1))$$
Now, the gauge potentials $\omega_i(q)$ are not external gauge fields; they are just functions on the configuration space; however, if they are shifted by a local $U(1)$ transformations, no physical prediction of the theory is altered. This is the reason why some call them fictitious.
The $\theta$ term in Yang-Mills theory (I'll just consider the Abelian case) is an example of the above theory:
After quantization, the electric field operator has the form:
$$E_i(x) = -i \frac{\delta}{\delta A_i(x)} - \frac{\theta}{8 \pi^2} \epsilon_{ijk}F_{jk}$$
Which can be written as:
$$E_i(x) = -i \frac{\delta}{\delta A_i(x)} - \Omega_i(\mathbf{A}(x))$$
with:
$$\Omega_i(\mathbf{A}(x)) = \frac{\theta}{8 \pi^2} \epsilon_{ijk}F_{jk}$$
We can check that $\Omega_i$ is a flat functional connection:
$$\frac{\delta\Omega_i }{\delta A_j(x)} - \frac{\delta\Omega_j }{\delta A_i(x)} =0$$
Thus, the theta vacua are in a one to one correspondence to
$$\mathrm{Map}(\pi_1(\mathcal{A}/\mathcal{G}), U(1))$$
Where $\mathcal{A}/\mathcal{G}$ is the space of gauge potentials-modulo gauge transformations.
Using the contractibility of the total space of gauge potentials and the long homotopy sequence, we have:
$\pi_k(\mathcal{A}/\mathcal{G}) = \pi_{k-1}(\mathcal{G})$
Thus, the classification is according to:
$$\mathrm{Map}(\pi_0(\mathcal{G}), U(1))$$
When the base space is $S^3$
$$\pi_0(\mathcal{G}) = \pi_0(\mathrm{Map}( S^3, U(1)) = \pi_3(U(1)) = 0$$
Which proves that the theta sectors in an Abelian theory are trivial.
Remark: The above description can be generalized to the non-Abelian case, in this case the classification of the flat bundles (thus, the nonequivalent representations) is according to the non-Abelian representations of the fundamental group. In this case, we get nontrivial inequivalent quantizations. This point was elaborated in my answer to the following question.
