It may help to realise that what we are talking about here is abstract algebra. This means that we are concentrating on the properties satisfied by "objects" and "operations" rather than the specific details of those objects and operations.
So here is an example: I am going to talk about the number $0$ and no other numbers at all, ever! (Very boring? - yes definitely.) Which of the axioms of an integral domain does this satisfy. Well for example, it satisfies the associative law since the only possible sums of three numbers we have are
$$0+(0+0)\quad\hbox{and}\quad (0+0)+0\ ,$$
and these are definitely equal. Another: is there a number $u$ such that $ux$ is always equal to $x$? Well, the only possible value of $x$ is $0$ - that's the only number we're talking about, remember? So all we need is $u0=0$, and that's true when $u=0$. So $0$ is the multiplicative identity, and we usually write the multiplicative identity as $1$. Remember, that doesn't mean it is $1$ in the usual sense - as I said above, we are not really paying attention to the specific details of our numbers. So in this case it is reasonable to say that $0=1$.
But as mentioned above, this is a really boring example and is not typical of other integral domains. That's why we choose to specifically exclude it in the definition.