At the end of the day, if you are storing inputs and outputs to a function as a pair of numbers - one for the actual value, and one for the derivative - and if addition and multiplication work the way you expect and propagate derivatives correctly - then you are using dual numbers, regardless of if you notate it a + b*h or {"value": a, "derivative": b}.
Pytorch does things slightly differently in that it is mostly focused on reverse-mode autodiff, and so it stores adjoints relative to the overall output rather than partial derivatives relative to the input, but this isn't really an entirely different thing, in the same way that the FFT isn't entirely different from the DFT.
There seems to be some confusion about the relationship between dual numbers and smooth infinitesimal analysis. Both have nilpotent elements, but with dual numbers the background logic is classical, whereas it isn't with smooth infinitesimal analysis.
EDIT: I see you've edited your post to try to get in some extra criticism after I've already responded. That's terrible form, so I'll just respond here.
Dual numbers are a nice way to get started with forward-mode autodiff, to which it is so related that the two are essentially the same thing with different labels. Pytorch instead uses reverse-mode autodiff. Reverse-mode and forward-mode autodiff are different, but not so different that they are entirely different things. Reverse-mode is, as I put it in my OP, "not much more advanced" than forward-mode, even if not identical.
What is entirely different, much more advanced, and what Pytorch really doesn't do, is anything like the "epsilon-delta proofs" you keep hanging your hat on. If Pytorch did that, it would be useless. The entire point of autodiff is to avoid such things.
Beyond that, I would suggest slowing down a bit as you are mixing quite a few things up. Nonstandard analysis has nothing to do with dual numbers at all, for instance. And you're very much misinterpreting that MSE post of mine you linked to (thanks!).