Why intersection is not an axiom in naive set theory? though, union was given as an axiom
The existence of intersection follows from the axiom schema of separation which informally tells you that if you start with a set and specify appropriately a way to select certain elements from this set by describing a formula that they satisfy then indeed there exists a subset containing precisely those elements. The construction of intersection fits this pattern (we can describe $A \cap B$ as selecting elements from $A$ that also lie in $B$) and so the existence of intersection follows from this axiom (for a rigorous proof, see Axiom schema of specification - Existence of intersection and set difference). The union construction generates from two sets $A,B$ a new set which is not a subset of either $A$ or $B$ in general and so requires another axiom to justify its existence.