There has been a lot done applying category theory to regular languages and automata. One starting point is the recent papers:
In the first of these papers, the structure of regular expressions is treated algebraically and the languages generated are dealt with coalgebraically. These two views are integrated in a bialgebraic setting. A bialgebra is an algebra-coalgebra pair with a suitable distributive law capturing the interplay between the syntactic terms (the regular expressions) and the computational behaviour (languages generated). The basis of this paper is algebra and coalgebra, as treated in computer science under the umbrellas of universal algebra and coalgebra, rather than what one sees in mathematics (groups etc).
The second paper uses techniques that come from the more traditional mathematical treatment of algebra (modules etc) and coalgebra, but I'm afraid that I don't know the details.
Neither treats Kleene star as an adjunction, as far as I can tell.
More generally, there is a lot of work applying category theory to automata instead of regular expressions. A sample of this work includes:
Bloom S.L.; Sabadini N.; Walters R.F.C. Matrices, Machines and Behaviors. Applied Categorical Structures, Volume 4, Number 4, December 1996 , pp. 343-360(18)
Michael A. Arbib, Ernest G. Manes: A categorist's view of automata and systems. Category Theory Applied to Computation and Control 1974: 51-64
M.A. Arbib and E.G. Manes. Adjoint machines, state-behaviour machines, and
duality. Journal of Pure and Applied Algebra, 6:313-344, 1975.
- M.A. Arbib and E.G. Manes. Machines in a category. Journal of Pure and Applied
Algebra, 19:9-20, 1980.
- Jirí Adámek and Vera Trnková's book Automata and Algebras in Categories, as pointed out in a comment.
Finally, there's the work on iteration theories, Iteration theories: the equational logic of iterative processes by Stephen L. Bloom and Zoltán Ésik, which focusses on iteration (e.g., Kleene star), but from a more general perspective, where regular languages are just one thing that falls under the theory.