The yoneda lemma gives us a characterization of $Psh(\mathcal{C})$ that seems very similar to the theory of distributions. That is, we have a notion of representable presheaves, similar to representable distributions. The ability to talk about presheaves as colimits of representables correlates to the more complicated notion of distributions as derivatives and limits of representables. The whole idea of "test objects" is exactly the same as the notion of "test functions" and so on. Is there a deep connection there or is it just another case of stretching the terminology?

While maybe not exactly what you were after, here is something that you might enjoy looking into, which relates presheaves and distributions. There exists a category of sheaves on certain test objects, such that
So in a way, this topos makes precise and manifest the intuition that distributions are "generalized functions". They are functions in this topos. The topos that I am talking about is described in great detail in section VI of the textbook Models for Smooth Infinitesimal Analysis. The test objects in this case, i.e. the objects in the site that the topos is a category of sheaves over, are smooth loci. Distributions are discussed in section VII,3 

