I got a strange idea a few years ago on my own. The story is here: if we have a measure space $(X,\Omega,\mu)$, where $\mu$ is non-negative, for its measurable subsets $A,B\subset X$, we can define a relation $$ A\sim B \Leftrightarrow \mu(A\Delta B)=0. $$ $\Delta$ means the symmetric difference. This is actually a equivalence relation, and if we denote $X^{\mu}$ to be the set of all measurable subsets of $X$, we get a new set $X^{\mu}/\sim$. Now, we can define a function $$d: X^{\mu}/\sim\to [0,\infty],$$ where $d(A,B)=\mu(A\Delta B)$. It is easy to verify that $d$ is a metric. Hence, we can give $X^{\mu}/\sim$ a metric space structure, and denote it $(X^{\prime},d)$. Thus we have an arrow $$(X,\Omega,\mu)\to (X^{\prime},d).$$ But if we want to define a functor, we have to find out how to get an arrow $$f: (X,\Omega,\mu) \to (Y,\Omega^{\prime},\mu^{\prime})\to f_*: (Y^{\prime},d_Y)\to (X^{\prime},d_X).$$ Here f is a measurable function. Therefore, there is a natural idea to define $f_*: B\mapsto f^{-1}(B)$. However, we need $f_*$ to be at least a continuous map (and well-defined).If we want $f_*$ to be well-defined, there must have $$f_*(D)=f_*(C), if\, C\sim D.$$ That is $$\mu^{\prime}(C\Delta D)=0\Rightarrow\mu(f^{-1}(C)\Delta f^{-1}(D))=\mu(f^{-1}(C\Delta D))=0.$$ To get continuity, we consider an open set $U=\{C\in X^{\prime}\vert d_X(C,A)< \epsilon\}\subset X^{\prime}$, $f_*^{-1}(U)$ should be an open subset of $Y^{\prime}$, that is, every point of $f_*^{-1}(U)$ is an interior point. Suppose $B\in f_*^{-1}(U)$, we have $ C=f^{-1}(B)\in U, i.e. d_X(C,A)< \epsilon$. To make $B$ an interior point, we must have a subset $V=\{D\in Y^{\prime}\vert d_Y(D,B)<\delta\}\subset f_*^{-1}(U)$ for some $\delta>0$. But this means that for all $D\in V$, $f_*(D)\in U, i.e., d_X(f^{-1}(D),A)<\epsilon$.
And here I have no good idea but restrict $f$ to be a function satisfying: $$\mu(f_*(B))\leq k\,\mu^{\prime}(B),\text{for some } k>0.$$ Now, I have defined a functor. Is this will be useful somewhere? or just a interesting nonsense?
I think one problem with using category theory to speak about calculus is that there are many somewhat canonical categories involved and it depends on the usecase, which one is "right".
In this specific instance there are for example at least two categories of measure spaces, which could be of interest. One is the category $\mathsf{Meas}_P$ of measure spaces $(X,\Omega,\mu)$ and measure preserving maps $f:(X,\Omega,\mu) \rightarrow (Y,\Lambda,\lambda)$, where $f$ is measurable and satisfies $\mu(f^{-1}(B))=\lambda(B)$ for $B\in\Lambda$.
Another choice could be the category $\mathsf{Meas}_{L}$, where morphisms are measurable functions satisfying a Lipschitz condition in the sense that there is $L>0$ such that $\mu(f^{-1}(B)) \leq L\lambda(B)$ for all $B\in\Lambda$. Note that there we also have $\lambda(B)=0\implies\mu(f^{-1}(B))=0$...
Similarly, there are different categories of metric spaces. One is the category $\mathsf{Met}_P$ of metric spaces (for me a metric takes values in $[0,\infty]$) and distance preserving maps. Another one is the category $\mathsf{Met}_L$ of metric spaces and Lipschitz maps.
Now your construction can be made into a functor $$\begin{array}{rcl} \mathsf{Meas}_P^\text{op} &\longrightarrow& \mathsf{Met}_P\\ (X,\Omega,\mu) & \mapsto & (\Omega/\sim,d)\\ f &\mapsto& f^{-1}/\sim \end{array}$$ It also gives rise to a functor $$\begin{array}{rcl} \mathsf{Meas}_{L}^\text{op} &\longrightarrow&\mathsf{Met}_L\\ (X,\Omega,\mu) & \mapsto & (\Omega/\sim,d)\\ f &\mapsto& f^{-1}/\sim \end{array}$$ since $$d(f^{-1}(B),f^{-1}(B'))=\mu(f^{-1}(B \Delta B')) \leq L \lambda(B\Delta B') = Ld(B,B').$$ Dropping the equivalence relation you could also define a functor $$\begin{array}{rcl} \mathsf{Meas}_{L}^\text{op} &\longrightarrow&\mathsf{PsMet}_L\\ (X,\Omega,\mu) & \mapsto & (\Omega,d)\\ f &\mapsto& f^{-1} \end{array}$$ where $\mathsf{PsMet}_L$ denotes pseudometric spaces and Lipschitz maps. Long story short, there are many different variants to consider. Which one is "correct" depends on what you intend to do with it. In any case this construction (especially the third functor) gives a clear answer to the question, in what sense one can regard a measure as a metric on a sigma-algebra.
I would like to end with a few followup questions. What categorical properties do these functors preserve? In particular, are they faithful or full? Does any of them have an adjoint functor? Is it possible to characterize those (pseudo)metric spaces lying in the essential image? How is the second functor related to the third functor (If I recall correctly $\mathsf{PsMet}_L \subseteq \mathsf{Met}_L$ admits a left-adjoint, so one could hope the second functor factors through that)?