No, such a connection cannot exist: Consider a function which vanish to first order at a point $p\in M$, i.e., $f(p)=0$ and $d_pf=0$, but assume that there are vector fields $X,Y$ with $(X\cdot Y\cdot f)(p)\neq0$.Such a function clearly exist. Moreover, let $\tilde Z$ be a vector field which does not vanish at $p.$ Then, the equality does not hold for the three vector fields $X,Y,Z=f\tilde Z.$

No, such a connection cannot exist: Consider a function which vanish to first order at a point $p\in M$, i.e., $f(p)=0$ and $d_pf=0$, but assume that there are vector fields $X,Y$ with $(X\cdot (Y\cdot f))(p)\neq0$.Such a function clearly exist. Moreover, let $\tilde Z$ be a vector field which does not vanish at $p.$ Then, the equality does not hold for the three vector fields $X,Y,Z=f\tilde Z.$