There are two kinds of adjunction. The easiest is when your field $K$ and your element $\alpha\
otin K$ are both sitting inside a bigger field $\Omega$. Then $K(\alpha)$ may be defined to be the smallest subfield of $\Omega$ that contains both $K$ and $\alpha$: it’s the intersection of all subfields $L\subset\Omega$ that themselves contain $K$ and $\alpha$, and there are other ways of describing it.
The other kind is sometimes called “abstract adjunction”, where you take a polynomial $f$ irreducible over $K$, and make up a new field $L$, defined to be $K[X]/(f(X))$, which contains a naturally isomorphic copy of $K$, and an element $\widetilde X$ that’s a root of $f$. This new field also has the property that nothing smaller than it contains both (the isomorphic copy of) $K$ and the root.
I don’t think that thinking of either of these processes as something like union would be at all helpful.