= Solution
Use <polarization identity> of the <quadratic variations>:
$$
\boxed{[M,N]=\frac14\bigl([M+N]-[M-N]\bigr).}
$$
This is continuous and adapted, starts at zero, and has <finite variation> on every compact interval because it is a difference of two nondecreasing processes. The defining <quadratic variation> property gives that $(M+N)^2-[M+N]$ and $(M-N)^2-[M-N]$ are <continuous local martingales>; their difference divided by four is $MN-[M,N]$. This proves existence without assuming a product formula involving an as yet undefined <quadratic covariation>.
If $C$ and $D$ both satisfy the requirements, $C-D$ is a continuous <finite-variation process> and a <local martingale>. By <continuous finite-variation local martingale is constant>, it is constant; its initial value is zero, so $C=D$ up to <indistinguishability of stochastic processes>.
There is a minor initial-value convention in this characterization. The displayed uncentred <local martingale> identity is directly valid when $M_0=N_0=0$, or when the initial terms have the requisite integrability. With arbitrary finite initial values, centre the <quadratic variation> construction and use $MN-M_0N_0-[M,N]$, which starts at zero. Adding back $M_0N_0$ requires it to be integrable under the usual definition of a <local martingale>; the <quadratic covariation> itself is unaffected by centring.
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