In a more abstract setting, "p is prime" means that if p|ab, then p|a or p|b, and "irreducible" means only divisible by itself or a unit (in this case 1). The FTA corresponds to unique factorization into irreducibles, and the fact that irreducible and prime are the same thing is a consequence of unique factorization. (In an integral domain, every prime is irreducible; in a unique factorization domain, the converse is also true).
... the standard terminology is confusing:
calling a number only divisible by 1 and
itself a "prime" already assumes the FTA.
But you are correct, and the reason we have these terms is exactly to avoid some of the "intuitively obvious" misconceptions.