A zoologist on reddit chimed in on how this is possible:
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Camel mouths are full of cone-shaped papillae that look like this. These protrusions are partly keratinised - keratin being the hard stuff your nails are made out of - which makes them tough n' semi-rigid, feeling a bit like the middle of tupperware lids when you squish 'em. The plastic-ey cones not only help protect the mouth from internal damage - scratches, abrasions etc. - when they feed on thorns and other nasties, but they also manipulate the food to go down in one direction.
Worth mentioning that modern camels wouldn't be eating cactus like this in the wild either; instead it'd be scrubby, thorny acacia bushes and the like. They also likely do feel some pain and discomfort eating this stuff, as much of their mouths - particularly their lips - are very sensitive, despite the papillae. Being metal as fuck though, camels just get on with it. They have an oddly voracious appetite for prickly pear and similar cacti native to North America, so clearly there's something about those plants that camels love, despite the irritating prickles. Makes them sort of sadomasochistic diners, really.
Anywho, the same sorts of papillae structures have independently evolved multiple times across the animal kingdom; notably inside the mouths and throats of leatherback turtles.