Obviously this is an open-ended question, so I'll be happy to try and clarify anything about it that doesn't make sense.
Let $X$ be a Riemann surface and $f$ be a meromorphic function on $X$. Of course a "meromorphic function" is simply a holomorphic mapping $X \to \mathbb{P}^1$. We can define the divisor associated to $f$ as
$(f) := f^{-1}(0) - f^{-1}(\infty)$
where those preimages are regarded as divisors in the usual way. However, viewed from this perspective, there's nothing special about the points $0$ and $\infty$ on $\mathbb{P}^1$ -- after all, $\operatorname{PSL}_2(\mathbb{C})$ acts 3-transitively on $\mathbb{P}^1$ -- so it seems like it would be more natural to associate to a function $f$ the map $(a, b) \mapsto f^{-1}(a) - f^{-1}(b)$, where the preimages denote divisors as above.
This situation is analogous to a potential in physics, where we have a quantity that's defined in a path-independent manner between any two points, but doesn't have a sensible definition at any given point. To carry this idea over, it'd probably make sense to topologize the divisor group. [1] Then we could think of the divisor of a function as a (holomorphic) map $(f) : \mathbb{P}^1 \times \mathbb{P}^1 \to {\rm Div} X$.
Does this line of thinking lead anywhere? Is it trivially equivalent to some other way of thinking about things? Does it generalize in any way to the case of, say, divisors on a scheme, where we can't interpret the sections $H^0(X, \mathscr{K})$ so simply?
[1] I'm not sure how to do this best, but for instance we could think of the effective divisors of degree $n$ as living in the configuration space of $n$ unordered points on $\mathbb{P}^1$ with repetition, $(\mathbb{P}^1)^n/\Sigma_n$. The case where we can have negative coefficients could then be handled by considering pairs of effective divisors modulo cancelling pairs. (I'm ignorant as to how nice or bad the resulting complex analytic space is, though.)