There are actually many circumstances in which we do want to treat 0 as a prime. For example it generates a prime ideal in the integers.
One way to state unique prime factorisation including 0 is to treat the set of primes as a pointed set with 0 as the point. The smash product makes the category of pointed sets into a symmetric monoidal category. Then unique prime factorisation can be stated by saying that the multiplicative monoid of natural numbers (with 0) is isomorphic to the free commutative monoid object on the set of primes.
One way to state unique prime factorisation including 0 is to treat the set of primes as a pointed set with 0 as the point. The smash product makes the category of pointed sets into a symmetric monoidal category. Then unique prime factorisation can be stated by saying that the multiplicative monoid of natural numbers (with 0) is isomorphic to the free commutative monoid object on the set of primes.