Pathwise uniqueness for a multiplicative martingale equation ($dZ=ZdM,Z_0=z>0⟹Z_t=z\exp (M_t-M_0-\frac{1}{2}[M{]}_t)$)

The Itô formula gives $d\log Z=dM-\frac{1}{2}d[M]$ for a strictly positive solution. Therefore equal initial values give indistinguishable solutions. Conversely the displayed exponential satisfies the equation. This is uniqueness for a fixed continuous local martingale driver and does not require the exponential to be a true martingale.