Note that D&F did not specify the existence of such an element, only that such an element, if it exists, satisfies certain properties.
As for your question, it is because factorisation may not be unique. In the integral domain , note that
Now and divides both and but neither can satisfy the second property of gcd , since does not divide or vice versa. Note that cannot be a gcd either because it divides both and , which are both factors of and and not the other way round.