> What does it mean to take a factorial on the real numbers, or to add only on the even integers?
I don't think that's exactly the same problem.
It is perfectly valid to define an operation on a particular domain. Defining an add operation on even numbers is perfectly fine (If you had said odd, well...). Factorial on real numbers is more or less the Gamma function. The problem here is that the operation is defined on one domain...
> We can say that division’s domain is all the real numbers except zero.
...but then, for some reason, it is decided that will will be applied on a different domain. That is, that we're going to apply on all real numbers including zero. So now we need to modify the definition of the operation. And the thing is that, the new definition, created to accommodate the new domain, usually tends to produce friction precisely on the new elements added to the domain. We excluded zero initially because that way the operation was defined in a simpler way. Now, having to consider zero implies that our definition becomes more "complex".
Now, we can change the definition in different ways. Is any one of those objectively "better" or "worse" than the rest? Well, that's what the discussion is all about.