Here's an alternative approach, which might allow using machinery from linear algebra. It's based on the sort of analysis used on Markov chains, and various details may well be wrong. The key idea is that the spectrum (multiset of eigenvalues) of the tensor product of two matrices is the multiset of pairwise products of each eigenvalue of the first matrix with each eigenvalue of the second matrix. This generalizes in the obvious way for a tensor product of $n$ matrices. Spectra are interesting, because the multiplicity of an eigenvalue $\lambda$ in the spectrum is (more or less) the dimension of the subspace of vectors which are sent to $\lambda$ times themselves when multiplied by the matrix.

Consider the adjacency matrix $G$ of the graph, but modify each row by dividing it by the total number of non-zero entries in the row. Rows that have no non-zero entries correspond to isolated vertices, and their entries remain zero. If we consider a vector $v$ as representing a probability distribution over the vertices, then $Gv$ is the probability distribution if the initial distribution "diffuses" one step -- the density at any one vertex diffuses in equal parts to all adjacent vertices (and none stays behind), where density at an isolated vertex goes away.

After playing with simple cases (the trivial graph, the minimal bipartite graph, cycles), I believe: (1) Each trivial component generates one eigenvalue $0$. (2) Each non-bipartite connected component with $k$ vertices generates one eigenvalue $1$ (representing the steady-state distribution on that component) and $k-1$ eigenvalues with absolute value between $0$ and $1$ that represent deviations from the steady-state distribution that "die out over time" in the diffusion model. (3) Each bipartite connected component with $k$ vertices generates one eigenvalue $1$ (representing the steady-state distribution), one eigenvalue $-1$ (representing the deviation from the steady-state distribution that oscillates stably between being one part and the other part of the bipartite component), and $k-2$ eigenvalues with absolute value between $0$ and $1$, representing decaying modes.

Now, looking at the spectrum of the tensor product $P$ of all the $G_i$ as the products of all combinations of eigenvalues of the $G_i$, taking one from each factor, we see -- where the number of connected components in $P$ is the sum of the multiplicities of $0$ and $1$:

$0$ eigenvectors of $P$ result when any factor eigenvalue is $0$. This corresponds to the enumerative result about the overpowering effects of a trivial connected component of a factor.

$1$ eigenvectors of $P$ result when all factor eigenvalues are $1$ or $-1$, and the number of $-1$ factors is even. This corresponds to the "signature" pairing in the enumerative result.

$-1$ eigenvectors of $P$ result from the same, except that the number of $-1$ factors is odd. Ditto correspondence.

All other combinations generate eigenvectors between $0$ and $1$, which do not represent connected components.