Jace - There were many happy accidents that went into the mix.
The first was accidentally NOT making them the way Seth Lover had intended. That included using machines for mass production not really set up for such small bobbins - machines winding each in their own way, and each of the (3 or 4) stations of each winder having its own idiosyncratic wind pattern. If you look at coils, they often have the wire bunched up on one end or the other, or with a dip in the middle and bunched on both flanges. The mixture of such variances and the way that coils were randomly chosen made for some incredible pickups.
Then you have the very inconsistent materials - the best of the day but still not very even. The wire and the loose tolerances of thickness of both copper and insulation matter. The poor purity of metals might also help explain the way modern mags don't quite sound the same, and things like screws and keeper bars help shape the magnetic field so composition matters here too.
The modern clone winders go through all of these aspects - and supposedly with pretty accurate results (I have played clones but not PAF's, so I can only take the word of those who have played both). But even then, with 2 sets of clones wound the same but 1 with the 1959 era wire and the other modern, there is still a difference in the 'depth' of the result.
Modern pickups that don't try and replicate any aspect of the old pickups have to do much more to 'compensate' for certain missing elements, but then having other artefacts appear. eg - If you want that nice biting top-end that vintage pickups have you might be tempted to use an A5 magnet - but you lose the fullness of the originals, or might make it a bit scooped in the process.