(A) Does anyone know some non-trivial algebraic structures $(S,\cdot,*)$ such that both $\cdot,*$ are associative, and with the property $$ x*(y\cdot z)=x*y*z\;\; ; \;\; \forall x,y,z\in S? $$ (B) Any such algebraic structures with $(S,\cdot)$ is a group?
(C) Any other similar algebraic structures?
Motivation. For every fixed real number $b\neq 0$, the structure $(\mathbb{R},+,+_b)$ satisfies $x+_by+_bz=x+_b(y+_bz)=x+_b(y+z)$, where $x+_by:=x+y-b[\frac{x+y}{b}]$ (see this , this and this).
Take any semigroup $(S, .)$ and define a new operation $*$ by setting $x * y = x$ for all $x, y \in S$. Then your conditions are satisfied. In particular, you can start with a group $(S, .)$.
EDIT. You can also take the quotient of the free term algebra (say on one generator) by the three identities $x*(y*z)=(x*y)*z$, $x\cdot(y\cdot z)=(x\cdot y)\cdot z$ and $x*(y\cdot z)=x*y*z$.