Can adjoint linear transformations be naturally realized as adjoint functors?

A neat correspondence between adjoint functions and adjoint functors is possible, if you relax your understanding of what it means for a category to "realize" a Hilbert space a bit. (The adjoint of a linear function only exists if the vector spaces are Hilbert spaces and the function is continuous, so I'll take the question to be about Hilbert space instead of vector spaces.)

Given a Hilbert space $H$, "realize" it as the partially ordered set of closed subspaces $S(H)$, regarded as a category. Then a continuous linear function $f \colon H \to K$ induces a contravariant functor $S(f) \colon S(H)^{\text{op}} \to S(K)$. Now, denoting the adjoint function of $f$ by $f^\dagger \colon K \to H$, we get an adjunction between $S(f)$ and $S(f^\dagger)$. In fact, up to a scalar, any contravariant adjunction between $S(H)$ and $S(K)$ comes from an adjoint pair of functions between $H$ and $K$!

All this comes from a 1974 paper by Paul H. Palmquist, a student of Mac Lane, called "Adjoint functors induced by adjoint linear transformations" in Proceedings of the AMS 44(2):251--254.

There's a canonical way of going the other way, starting with two linear categories with nice finiteness properties, with adjoint functors between them and getting a pair of vector spaces with adjoint linear transformations. The vector spaces are generated by formal symbols for each object in the category, and the inner product between any objects is the dimension of the Hom space (so Hom spaces had better be finite dimensional). Note that this doesn't have to be symmetric.

Functors give linear transformations, and adjoint functors are adjoint in the usual sense.

You can soup up this construction when you have some more structures on your category. For example, if you have a direct sum, then you can impose the relation $[A+B]=[A]+[B]$, and everything will work fine.

If your category is abelian, you can take Grothendieck group, where $[A]+[C]=[B]$ for every short exact sequence $0\to A \to B \to C\to 0$, but then you have to be much more careful about the fact that lots of functors (including Hom with objects in the category!) aren't exact: they don't send short exact sequences to short exact sequences. You need to use derived functors to fix this.

There's no canonical way of going the direction you asked, though in practice we have a very good record of being able to and I don't know of any really good examples of there being two equal natural seeming but different such constructions.

I think it's more natural to take advantage of the monoidal structure and regard the vector spaces as functors rather than objects. For simplicity, consider only finite dimensional vector spaces. Given V, we have a functor $F_V: Vect \to Vect$ which sends $W$ to $W \otimes V$. The familiar identification $Hom(U\otimes V, W) = Hom(U, W\otimes V^*)$ shows that the (category theory) adjoint of $F_V$ is $F_{V^*}$. (That's $F$ sub $V^*$, in case the font is too small to read.) Chaining together two of these adjunctive identifications of Hom sets, we have

$Hom(V, X) = Hom(1, X\otimes V^*) = Hom(X^*, V^*)$.

The above identification sends a linear transformation $g:V\to X$ to the (linear algebra) adjoint $g^*: X^*\to V^*$. If $V$ and $X$ are inner product spaces then we can of course identify $V^*$ with $V$ and $X^*$ with $X$.

Maybe that's too elementary and not the answer you were looking for. But it seems to me it's the most simple and obvious way to relate linear algebra adjoints to category theory adjoints.