If you interpret the Myhill-Nerode theorem as being about the algebraic structure of the corresponding family of automata and the monoid associated with that family, then there is an interesting necessary and sufficient generalization to context-free languages. One version of this is here -- http://arxiv.org/abs/math.GR/0601061. It is, in effect, that a language $L$ is context free iff it is accepted by a polycyclic monoid automaton iff it is accepted by a free group automaton.
For both these kinds of automata, the monoid or group acts as secondary storage attached to a non-deterministic finite-state control mechanism. Transitions in secondary storage are accomplished by right multiplication in the monoid or group, and the automaton starts with the identity 1 in the store and accepts only if it ends in an accepting state with 1 again in the store. The finite-state control cannot directly "read" the monoid/group store, but it essentially blocks based on what happens in the store, and that, combined with the non-determinism in the finite-state control, confers a lot of computational power on the combination.
The complete state set for the resulting automaton is thus $Q \times M$ where $Q$ is the state set of the finite state control and $M$ is the monoid (or group). For regular languages, the state set is just $Q$, which is finite, or $Q \times M$ where $M$ is just $\{1\}$ or finite. For CFLs $M$ has the structure stated above. So the generalization is about the relationship between the family of languages and the structure of $M$.
To finish the story, a polycyclic monoid is strings over a finite set of symbols $X^' = \{L_x, R_x: x \in X \}$ for some finite set of symbols $X$. The size of $X$ is the rank of the polycyclic monoid. Multiplication in this monoid is given by: $L_x R_x = 1$, $x \neq y \Rightarrow L_x R_y = 0 $, anything else just freely concatenates the right-hand symbol. Think of $L_x$ a pushing x onto a stack (right hand end of a string), $R_x$ as popping x. Trying to pop a symbol that isn't on top blocks the computation. So a polycyclic monoid is mimicking a pushdown store with k symbols, and it's not hard to see that automata using it as a store define the context-free languages.
That makes it seem that we've just trivially algebraicized the intuitive notion of a pushdown store, and indeed we have. But a polycyclic monoid is definable in a number of other ways, for example as the free group on k symbols, with $R_x$ acting as $L_x^{-1}$, and then things get more interesting -- Context-free-ness and group-ness are deeply related, as dtldarek points out, and the finiteness of the whole automaton for regular languages finds an analogy in the finite index of the free group.