All four of those factors are prime in the new number ring, giving 6 a dual existence
Those factors are irreducibles, but they aren't primes, which is the reason why the uniqueness of factorization fails.
A nonzero non unit element of a ring is called prime if p|ab implies p|a or p|b.
A nonzero non unit element of a ring is called irreducible if p=ab implies that a is a unit or b is a unit (invertible element).
Primes are irreducibles in an integral domain, but the converse is true in unique factorization domains and Z[√-5] is not one.