Why is Zero not composite?

The answer they're looking for, I think, is that if $1$ is considered a prime or $0$ considered a composite, then we no longer have unique factorization. Or at least the statement of the unique factorization theorem becomes uglier.

For the first point, $6 = 2\cdot 3 = 1^5\cdot 2 \cdot 3$ gives two different prime factorizations of $6$. Ick.

For the second, $0 = 0*3 = 0 *2.$ So $3$ is a factor of the first product, but not the second. Again, ick. And this breaks even more things. What's gcd$(0,0)$, for instance?

While $0$ is indeed divisible by any prime, no product of primes will make $0$. Therefore, $0$ is not composite.

We can of course assume that $0$ is a composite number. It would not break mathematics or anything. However it would make a lot of theorems and statements more tedious. For instance, we know that any composite number can be written as a product of a finite amount of primes. However If we let $0$ be composite then we have to either always say "Every composite except $0$ can be ..." or ignore such theorems.

To put it simply we let $0$ be non-composite because it is convenient.

One could also argue that "composite" numbers multiplied with each other should create a new composite number, and this is not true for $0$. But then this is more of a philosophical stance than a mathematical one I would claim.