Where does the proof of unique factorization fail for $\mathbb Z[\sqrt{-5}]$?

Question

I know that unique factorization does not hold for all rings, such has the much-used example $\mathbb Z[\sqrt{-5}]$. It seems that Euclid's lemma does not hold for these rings, and so on. However, there are proofs of the fundamental theorem of arithmetic which do not use Euclid's lemma, and moreover do not seem to use any particular properties of the positive integers (see for instance this or (better) the alternate proof given here). Why does this proof fail for $\mathbb Z[\sqrt{-5}]$ and similar?

Answer 1

This answers the alternate proof in your second link: It starts off using the well ordering principle of the natural numbers. But we don't have that basic well ordering of our usual $<$ on $\mathbb Z [\sqrt {-5}]$ that cooperates nicely with our other operations.

Answer 2

Intuitively the cell of the lattice built with $1$ and $\sqrt{-5}$ is too large. The counterexample is $6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})$