I doubt this is something essential to type theory per se; but more due to the pervasive mode of mathematical and scientific writing; the art of exposition, I think has been gradually lost.
Lagrange for example prided himself that his mathematical works had no diagrams. Vladimir Arnold had complained about an excessive Bourbakiste approach to mathematics that instead of clarifying the main ideas and sign-posting significance rather hid everything under a thick encrustation of formalism; most of which hid that little of any additional significance was being said.
Certain type theories are related to certain categories - for example the simply typed lambda calculus to closed Cartesian categories and some of these can be presented more naturally as string diagrams; and these can be understood topologically in a natural way.
I'd argue this is a kind of diagrammatic analogy.
In Set Theory, elements and sets are basic; and the notion of a function is derived; Topos Theory takes sets and functions (ie objects and morphisms) as basic; and elements as derived: Lawvere said he wanted to do set theory without elements.
This isn't quite a metaphor, or even an analogy but it works quite well in the way it turns Set Theory 'on its head'; rather like how Marxism as economic philosophy turned Hegel 'on his head'. One might call it a slogan; perhaps even a'mission statement'.