I am wondering if there is a nice categorical description of matrix similarity, i.e. the equivalence relation on matrices given by $A\sim B \iff A=QBQ^{-1}$, for some invertible $Q$.
In particular, I am considering the matrices as linear maps from a vector space into itself, which in turn forms a category $\mathrm{End}(V)$ with one object $V$ and arrows linear maps. We can then form the functor category $\mathrm{End}(V)^{\mathbb2},$ where $\mathbb2$ is the category with $2$ objects and $1$ arrow between them, i.e. this is the 2-category of linear maps and pairs of maps between them making the following commute:
\begin{array}{ccccccccc} V & \xrightarrow{A} & V \\ \downarrow & & \downarrow \\ V & \xrightarrow{B} & V \end{array}
So similarity between $A$ and $B$ is stronger than isomorphism in this category, as $(C,D):A\rightarrow B$ is an isomorphism if and only if $C$ and $D$ are linear isomorphisms, whereas similarity would require $C=D$. Do similar matrices in this category obey some characterising or otherwise interesting property, or is isomorphism the strongest we can get?
$\DeclareMathOperator\rank{rank}$If you allow $C\neq D$, an isomorphism $A\cong B$ in the category you describe is equivalent to $\rank(A)=\rank(B)$. You can of course define a category where as morphisms you only allow commutative squares with $C=D$, then an isomorphism $A\cong B$ is indeed equivalent to $A$ and $B$ being similar.
Another way you might want to look at this is in the setting of $K[X]$-modules, where $K$ is the underlying field. A $K[X]$-module amounts to a $K$-vector space $V$ together with a $K$-linear map $f\colon V\to V$ that describes the action of $X$, that is $f(v) = X\cdot v$. In this setting, matrices $A$ and $B$ yield $K[X]$-modules and those are isomorphic if and only if $A$ and $B$ are similar.