In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof bypasses Euclid's lemma to prove that $ \mathbb{Z}$ is a UFD. It seems to me that the deepest property of the ring of integers used is some properties of the order on $ \mathbb{Z}$! An account of this proof can be seen here:

https://planetmath.org/inductionproofoffundamentaltheoremofarithmetic

Can this idea be applied or has it been applied to any other situations? Or is it simply a tricky proof for the ring of integers?