  In this work we prove uniqueness of distributional solutions to $2D$
Navier-Stokes equations in vorticity form $u_t-\nu\Delta u+ div (K(u)u)=0$ on
$(0,\infty)\times\mathbb{R}^2$ with Radon measures as initial data, where $K$
is the Biot-Savart operator in 2-D. As a consequence, one gets the uniqueness
of probabilistically weak solutions to the corresponding McKean-Vlasov
stochastic differential equations. It is also proved that for initial
conditions with density in $L^4$ these solutions are strong, so can be written
as a functional of the Wiener process, and that pathwise uniqueness holds in
the class of weak solutions, whose time marginal law densities are in
$L^{\frac43}$ in space-time. In particular, one derives a stochastic
representation of the vorticity $u$ of the fluid flow in terms of a solution to
the McKean-Vlasov SDE. Finally, it is proved that the family
$\mathbb{P}_{s,\zeta},$ $s \geq 0$, $\zeta=$probability measure on
$\mathbb{R}^d$, of path laws of the solutions to the McKean-Vlasov SDE, started
with $\zeta$ at $s$, form a nonlinear Markov process in the sense of McKean.
