Something can be exponential in one context and polynomial in another. Asymptotic analysis is about the growth rate of a function in terms of some input variable. The results you get depend on which values you assume to be fixed while others vary. It is usually applied to analyze the runtime of a program in terms of the input size, which is the basis of classifying the runtime complexity, but that's not the only way you can do it.

If you have a sequence of programs with polynomial runtime, but which grows as N, N^2, N^3, N^4, ..., that's a textbook example of exponential growth. It is not generated by running a program on increasingly larger inputs, so there's nothing to be put in the exponential runtime complexity class, but it's exponential nonetheless.

In this case, the number of iterations (1000) of each loop is being held fixed, and the depth of nesting is varying, i.e. the body of the inner loop executes O(1000^depth) times.

OK I had to write things down to try and make sense of this. If anyone is like me, consider this loop that loops 5 times... maybe it will help?

  defines = 0
  tests = 0
  increments = 0

  defines++
  for (let i = 0; i < 5; i++) {
    tests++
    increments++
  }

  console.log(`defines: ${defines}, tests: ${tests}, increments: ${increments}`)

For the sake of space I'm not going too copy nested versions of that, but imagine nesting it two and then three levels deep.

  1 level will increment 5 (5^1) times    
  2 levels will increment 25 (5^2) times    
  3 levels will increment 125 (5^3) times

More interesting are the number of definitions and tests:

  levels    tests    defines
  1         15       1 
  2         30       6
  3         155      31

But I don't know is how the interpreter works and whether it has tricks to shrink those numbers; those just reflect my naive mental model of how things work.

If N is the number of nested loops, and M is the number of times through the loop, then it is indeed a O(M^N). So indeed, complexity scales exponentially with the level of nesting. The wording was just off amd confusing due to it being a nontraditonal formulation of the problem, but what he was saying does actually make sense.