The point of talking about topoi is rarely to prove something new or interesting about a particular category of spaces. Quite the opposite, the point is typically to prove something interesting about topoi which applies to all categories of spaces that also happen to be topoi. The point is that the abstraction is powerful enough to prove quite a bit for many interesting categories.

Categories of Banach ($\textbf{Ban}$ of all and $\textbf{Ban}_{fd}$ of finite dimensional) and Hilbert spaces ($\textbf{Hilb}_{\oplus}$, $(\textbf{Hilb}_{\oplus})_{fd}$, $\textbf{Hilb}_{\otimes}$, and $(\textbf{Hilb}_{\otimes})_{fd}$) (and related spaces of interest to analysis) can be seen as symmetric monoidal categories, with additional structure. Like topoi, monoidal categories are important and powerful structures with many important theorems proven in them.

These categories are Galois adjoint to topoi in ways that allow natural internal logics to be built and find consonance with the adjunction. This is one way in which topoi are seen in relation to these symmetric closed monoidal categories. You see this association, for instance, in operationalist reductions of quantum theories (which are formulated over a category of Hilbert spaces).

But although these adjoints have some structure-preserving properties, I have found that many treatments stick with the natural categorial structure and prove things in the natural internal logics. Braided monoidal categories have a huge amount of structure and really, I think many will agree that two of the most important types of categories to study when learning categorial methods are topoi and monoidal categories, as that already gets a good part of modern mathematics.

However, if you are interested in topos theoretical direction exclusively, another direction important in analysis besides that above and those of other answers is Pestov's Conjecture, that there exists a Grothendieck topos whose internal Banach spaces are operator spaces. This direction of research is intended to provide a natural category that shows which theorems on Banach spaces extend to operator spaces in general.