Comment by cvoss

The connection between type isomorphisms and algebraic equalities is known and explored in a variety of interesting ways.

I don't know the precise history of the development of the notation and vocabulary, but there's a reason types like A+B are notated with a plus symbol and called "sums". Similarly for products. And it may surprise you (or not) to learn that people sometimes call A -> B an exponential type, and even notate it with superscript notation.

One of my favorites is the result that the binary tree type is isomorphic to septuples of binary trees, which you can arrive at by using the definition of binary trees as the fixed point solution of T = 1 + T*T and deriving T = T^7 with some algebra. [0]

[0] https://arxiv.org/abs/math/9405205