What is meant by a "shape" in programming

The way data is structured/organized defines it's "shape".

TypeScript specifically has type aliases so you may see something like:

type RefreshFn = () => boolean;

type Source<T> = {
  value: T | undefined;
  sinks: Set<Sink>;
};

type Sink = {
  sources: Set<Source<unknown>>;
  refresh: RefreshFn;
  isStale: boolean;
}

So in terms of TypeScript Sink is viewed as a "type" - which defines it's shape - the way the data is structured/organized (rather than viewing it as an object literal); an isStale property with a boolean value, a refresh property with a function that returns a boolean and a sources property that nests zero or more Source data structure in a Set.

Now given that JavaScript is dynamically typed one might refer to the "shape" of a Sink instance when talking about how it organizes its data.

Given Duck Typing in JavaScript one may state that two instances of data have a "similar shape" if they both include all the pieces of data needed to perform the operation (function) in question while either of them may have potentially different additional pieces of data that are irrelevant to the operation (function).

Now the term "shape" isn't limited to JavaScript but applies anywhere where data can have structure/organization but I find it is more often used in reference to dynamically typed languages. It's also used more in circumstances where data and behaviour are separated - i.e. outside of class-based object orientation.

So while Rust has struct, JavaScript uses object literals to emulate structured data and both constructs are used to "shape" the data.

Maybe it's to do with the structure of data, like how deeply nested a JSON object's properties go...

I've never heard that expression. Do you have a concrete example? I've heard design of the code (often related to en.wikipedia.org/wiki/Software_des...) or structure or data. But not "shape".