Literally everything I do, see, and experience feels mathematical in some sense, and I am completely incapable of turning it off. All I see are geometric objects with their various symmetries and topological properties, quantities of different objects, the PDEs associated with the changes of physical systems, and on and on and on.

Are you String Theorist or Supergravity Theorist ?

In addition, I feel like I'm perpetually thinking at like 15 different levels of abstraction simultaneously, and I'm constantly noticing the analogies (and their associated homomorphisms) present in my very thought processes. This has led to many interesting interactions with other people, since I usually feel like there are several different subjects in the conversation at once.

O.O, all I can do is play around with objects in a imagery universe(Complex Analysis(Undergraduate))but I want to get to that level where I see the world as an abstract landscape it seems you a very good mastery of Analysis and Abstract Algebra allowing to see deeply within a certain structure and see how that particular structure links with other structures.