  We characterise the multiplicative chaos measure $\mathcal{M}$ associated to
planar Brownian motion introduced in [BBK94,AHS20,Jeg20a] by showing that it is
the only random Borel measure satisfying a list of natural properties. These
properties only serve to fix the average value of the measure and to express a
spatial Markov property. As a consequence of our characterisation, we establish
the scaling limit of the set of thick points of planar simple random walk,
stopped at the first exit time of a domain, by showing the weak convergence
towards $\mathcal{M}$ of the point measure associated to the thick points. In
particular, we obtain the convergence of the appropriately normalised number of
thick points of random walk to a nondegenerate random variable. The normalising
constant is different from that of the Gaussian free field, as conjectured in
[Jeg20b]. These results cover the entire subcritical regime.
  A key new idea for this characterisation is to introduce measures describing
the intersection between different Brownian trajectories and how they interact
to create thick points.
