I think it is actually bad notation. It was a long time ago but I think the suggestive nod was basically pointing out how the entries of the matrix are arranged as "copies of del" and that this thing should be thought of as eating 2 infinitesimal vectors. I don't think he actually wrote that (again, the prereqs didn't make sense to try to do so), but afaik some people do use that notation.

I think if you tried to actually make sense of it, you'd expect that you plug in f twice (I think ∇ already secretly lives in R^n* ⊗ L(C^infty, C^infty), so you'd expect 2 copies of each of those). If you think of it as ∇: L(C^infty, L(R^n, C^infty)) then the composition ∇^2: L(C^infty, L(R^n, L(R^n, C^infty))) ≅ C^infty* ⊗ R^n* ⊗ R^n* ⊗ C^infty, so the types should work correctly (though the type of ∇ actually needs some generic parameters for composition to make sense), and you get that you only plug f in once.

Unfortunately ∇^2 is already taken by the Laplacian, so I suppose ∇⊗∇ would be "∇^2, but the one that makes a tensor", and it'll do as long as you don't try to type check it, which physicists won't.