I’m not sure how they’re doing it but in Germany all those PET bottles go into a centrally-managed recycling stream (because 25ct deposit) and I bet they have some technical norms around that kind of stuff. The bottles are all crushed to save space, incl. the caps, which at least in the case of the water bottle next to me is HDPE. Judging by the haptics the label is PET, a flimsy banderole glued (fused?) on at the seam.
Either they’re doing it chemically by breaking up the PET and then fishing out the rest from the soup (is that possible?) or what would also work I guess is shredding and mechanical sorting – the label is flimsy, the bottle always transparent, the cap never transparent. Such stuff.
Isn’t that what tapes are for.