Is this generalization of context free grammars known and strict?
I think the answer to your question can be found in Theorem 3.1 of [1], where it is credited to Wechler [2]. An algebra is stable if it is closed under left quotient.


  A language is context-free iff it belongs to a finitely generated
  stable algebra.


[1] Berstel, J.; Boasson, L. Towards an algebraic theory of context-free languages. Fund. Inform. 25 (1996), no. 3-4, 217--239. 

[2] Wechler, Wolfgang. Characterization of rational and algebraic power series. RAIRO Inform. Théor. 17 (1983), no. 1, 3-11.