The part you're missing is that the solution includes the solver gifting a camel to make 18, then taxing back the leftover camel at the end.

Of course, that's still got the same kind of 'but that's not in the rules' issue, but I think the bigger element is that it's not really meant as a "puzzle", but more showing off the unintuitive nature of the results.

That's why it's a puzzle and not a textbook word problem. Though maybe word problems should be puzzles more often so that students don't just plug the numbers into a formula and report the result without thinking whether it would be a good solution in the real world.

Note that the integer solution leaves no son cheated out of their inheritance. Everyone gets their apportionment, and a little more.

In addition to the observations in the sibling comments, note that "one half of the camels should go to the eldest heir, one third of the camels should go to the second heir, and one ninth of the camels should go to the youngest heir." -> 1/2 + 1/3 + 1/9 = 9/18 + 6/18 + 2/18 = 17/18ths, not 18/18ths. The will "happens" to be incorrect and fails to allot the full inheritance in exactly the way it needs to be for this to work.

There's a Wikipedia page about it: https://en.wikipedia.org/wiki/17-animal_inheritance_puzzle

You are correct that it's not solvable in a rigorous mathematical sense as stated.