Category-Theoretic relation between Orbit-Stabilizer and Rank-Nullity Theorems

The intuition behind this question is spot-on. I'm going to try to fill out some of the details to make this work.

The first thing to note is that a linear map $A:V\to V$ also gives a genuine group action: it is the additive group of $V$ acting on the set $V$ by addition. That is, any $v\in V$ acts on $x\in V$ as $v: x \mapsto x+Av.$

Now we see that given any $x$ in $V$ the stabilizer subgroup $\text{stab}(x)$ of this action is precisely the kernel of $A.$ The orbit of $x$ is $x$ plus the image of $A.$

If we are working with a vector space over a finite field, we can take the cardinality of these sets as in the formula $|\text{orb}(x)||\text{stab}(x)| = |G|$ and as @Ravi suggests, take the logarithm of this where the base is the size of the field and we get exactly the rank-nullity equation.

If we have an infinite field then this doesn't work and we need to think more along the lines of a categorified orbit-stabilizer theorem. In this case, for each $x\in V$ we can find a bijection:

$$ \text{orb}(x) \cong G / \text{stab}(x) $$

and as @Nick points out, this bijection gives us the First Isomorphism Theorem: $$ \mathrm{Im}(A) \cong V / \mathrm{Ker}(A). $$