  We consider a stochastic nonlinear defocusing Schr\"{o}dinger equation with
zero-order linear damping, where the stochastic forcing term is given by a
combination of a linear multiplicative noise in the Stratonovich form and a
nonlinear noise in the It\^o form. We work at the same time on compact
Riemannian manifolds without boundary and on relatively compact smooth domains
with either the Dirichlet or the Neumann boundary conditions, always in
dimension 2. We construct a martingale solution using a modified
Faedo-Galerkin's method, following arXiv:1707.05610. Then by means of the
Strichartz estimates deduced from arXiv:math/0609455 but modified for our
stochastic setting we show the pathwise uniqueness of solutions. Finally, we
prove the existence of an invariant measure by means of a version of the
Krylov-Bogoliubov method, which involves the weak topology, as proposed by
Maslowski and Seidler. This is the first result of this type for stochastic NLS
on compact Riemannian manifolds without boundary and on relatively compact
smooth domains even for an additive noise. Some remarks on the uniqueness in a
particular case are provided as well.
