The von Staudt-Clausen theorem indeed asserts that the denominators of Bernoulli numbers have special behaviour, but from a modern perspective what is even deeper are the so-called Kummer's congruences , whose proof normally relies upon von Staudt-Clausen.

What these congruences say is that evaluating the Riemann $\zeta$ function at integers that are $p$-adically close to each other, gives you integers which are again $p$-adically closed. In other terms, this asserts that if you want to define a function $\zeta_p:\mathbb{Z}_p\to\mathbb{Z}_p$ simply by insisting that it takes the same value of the usual Riemann $\zeta$ function on the negative integers, you can do this because it will be continuous (by Kummer) and then it will be unique, since $\mathbb{Z}_{<0}$ is dense in $\mathbb{Z}_p$.

The generalization you are looking for can then be restated as asking whether for each Dirichlet character $\chi$ we can find a *continuous* function
$$
L_p(\chi,s):\mathbb{Z}_p\to\mathbb{Z}_p
$$
which takes the same values as the complex function $L(\chi,s)$ when evaluated in a dense subset of $\mathbb{Z}_p$ consisting of integers (so that it make sense to compare the two functions). The answer is indeed affirmative, and this function is the so-called Kubota-Leopoldt $p$-adic $L$-function (eventually re-constructed by Iwasawa and Coleman). An extremely detailed, and still accessible, description of this is contained in Chapter 5 of Washington's *Introduction to Cyclotomic Fields*.

Indeed, much more is true. As BY points out in her/his comment, Deligne-Ribet (their unique joint work, I guess) and Cassou-Noguès (*Inventiones Math.*, 1979) and Barsky proved the existence of a $p$-adic gadget $\zeta_{p,F}:\mathbb{Z}_p\to\mathbb{Z}_p$ which "interpolates" $p$-adically the special values of the Dedekind $\zeta_F$ function attached to any totally real number field $F$, and a similar statement was proved by Katz when replacing $F$ by a CM field. Both Deligne-Ribet's and Cassou-Noguès' or Barsky's proof of the existence of such a continous function heavily realies on proving some *congruences* satisfied by the special values of the complex functions under consideration, precisely in the same spirit as Kummer's congruences were crucial (or equivalent) to prove the continuity of the Kubota-Leopoldt $p$-adic $L$-function. These congruences are usually referred to as ``Coates' congruences'' and were stated by John Coates in the paper *$p$-adic $L$-functions and Iwasawa theory* appeared in the proceedings *Algebraic Number Fields* edited by A. Fröhlich in 1977 (and published by Academic Press).

Then, the whole theory saw a striking explosion, providing for $p$-adic analytic function interpolating the complex $L$-functions attached to many arithmetic objects, and you can look for instance at the complex $L$-function attached to an elliptic curve or, more generally, to a modular form. Such a $p$-adic analytic object indeed exists (it is the so-called Amice-Vélu,Visik,Mazur-Tate-Teitelbaum $p$-adic $L$-function) and you can regard its continuity as a generalization of Kummer's congruences and, hence, of von Staudt-Clausen. The literature is huge, but you can start by reading the comments at the end of Chapter 5 in Washington's book for some hints.