Let me give an answer from a slightly different point of view.
Let $M_k$ be a moduli space as in your question; say it's a (compact) moduli space of sheaves on some (compact) Calabi-Yau threefold. In general, $M_k$ is going to be very singular. However, it carries a so-called perfect deformation-obstruction theory of dimension zero. This gives a virtual fundamental class on $M_k$, and the technical definition of the numerical invariant $e(M_k)$ is that it is the degree of this virtual fundamental class.
In the case of sheaves on CY3s, the deformation-obstruction theory has a duality property: it is a symmetric obstruction theory. In this case, according to a result of Kai Behrend, $e(M_k)$ can also be expressed as an Euler characteristic, albeit a weighted one:
$e(M_k)=\chi(M_k, \nu_{M_k})$,
where $\nu_{M_k}$ is the Behrend function of the singular space $M_k$. In other words, one computes an Euler characteristic, but weighted with a numerical measure of how bad the singularities are.
On can hope that this Euler characteristic definition can now be turned into something motivic. What one needs is a way to attach a motivic weight to points of $M_k$. In some specific moduli problems, such as for Hilbert schemes of points where at least locally the moduli space can be expressed as a critical locus of a function on a smooth variety, this can be done using the tool of the motivic vanishing cycle; indeed, this is what our work does in the paper you cite. The general theory of how one attaches motivic weights is discussed in a (partially conjectural) paper of Kontsevich and Soibelman.
The issue with Gromov-Witten theory on a CY3 is that the deformation-obstruction theory in that case, while it is of dimension zero, is not fully symmetric. It is symmetric on the open part corresponding to stable maps which are immersions from a smooth curve, but (as an expert assures me) not on the whole moduli space.
After thinking, and reading other references and re-reading the papers I mentioned, I may have found a sufficient explanation (at least to my care): Both instantons/monopoles are solutions to corresponding equations of motions from associated actions, and they "bloom" from an overarching SUSY action.
Witten formulated "twisted N=2 Supersymmetric Yang-Mills", a TQFT with SUSY (supersymmetry), which leads to the Donaldson invariants. This used an $SU(2)$-bundle over $X$ along with a gauge field (connection $\omega$) and matter fields (bosonic $\phi,\lambda$ and fermionic $\eta,\psi,\zeta$), and gave the Donaldson-Witten action functional $S_{DW}=\int_Xtr(\mathcal{L})$,
$\mathcal{L}=\frac{1}{4}F_\omega\wedge(\ast F_\omega+F_\omega)-\frac{1}{2}\zeta\wedge[\zeta,\phi]+id^\omega\psi\wedge\zeta-2i[\psi,\ast\psi]\lambda+i\phi d^\omega{\ast d^\omega}\lambda-\psi\wedge\ast d^\omega\eta$.
This has associated partition function $Z_{DW}=\int e^{-S_{DW}/g^2}D\Phi$ (here $\Phi$ denotes the space of aforementioned fields), where $g$ is a coupling constant that is the key here for answering our question. The "blooming" of this action functional is beyond the scope of my intentions and probably of MathOverflow, so I won't question it.
In weak coupling ($g\rightarrow 0$, known to physicists as the ultraviolet region), the action localizes to the classical Yang-Mills $S_{YM}=\int_Xtr(F_\omega\wedge\ast F_\omega)$ and have the equations of motion $d^\omega\ast F_\omega=0$. The global-minima solutions are $F_\omega=\pm\ast F_\omega$ (as Oliver clarifies in a comment). These solutions are the Donaldson instantons.
Now apparently, when we instead look at strong-coupling ($g\rightarrow\infty$, known to physicists as the infrared-region), the Seiberg-Witten equations should arise (a "duality" in Witten's TQFT). Indeed, Seiberg and Witten showed that this infrared limit of the above theory is equivalent to a weakly-coupled $U(1)$-gauge theory (the $SU(2)$-gauge group is spontaneously broken down to the maximal torus). Perhaps here is where a better understanding would be desirable (buzzwords 'asymptotic freedom' and 'symmetry breaking' appear).
Anyway, some physics-technique stuff happens (the previous paragraph can be described as "condensation of monopoles"), and we must consider a spin-c structure (which all of our oriented 4-manifolds have, whereas a spin structure would not allow us to consider all 4-manifolds); note that $Spin^c(4)=(SU(2)\times SU(2))\times_{\mathbb{Z}_2} U(1)$. This gives the data: $U(1)$-gauge field $A$ and positive spinor field $\psi$ (as written in the original post). The pair $(A,\psi)$ is a monopole when it minimizes an action $S_{SW}$, i.e. are time-independent solutions to equations of motions (the Seiberg-Witten equations). The action here is $S_{SW}=\int_X(|d^A\psi|^2+|F_A^+|^2+\frac{R}{4}|\psi|^2+\frac{1}{8}|\psi|^4)$, with scalar curvature $R$.
I hope this post is not too confusing.
[[Edit/Update]]: I just came across a book chapter by Siye Wu, The Geometry and Physics of the Seiberg-Witten Equations. These lectures tell the physical origin completely! (i.e. completely details my sketch). The SW equations and action functional pops up on pg191.
http://www.springerlink.com/content/q37322037j466218/
Best Answer
In 8 dimensional case we do not have direct analogous theory like 4 dimension case by strong weak duality. One can naively consturct the SW theory for CY4(line bundles with sections), but the theory turns out to be quite trivial since we need the virtual dimention of the moduli space is topological and we do not have much choice. In 4 dimensions, it is Kroheimer and Mrowka first showed that Donaldson polynomials have recurrence relations for simple type 4 mfds. Then Seiberg and Witten wanted to understand this from Physical perspective and finally got to SW theory. All Gauge theories in 4 dim are expected to be recovered by SW theory. But this is far from clear even for CY3(people seems only consider DT invs for curves(=GW by MNOP) and pts(computed by several groups) so far).