  In this paper, we investigate a class of planar Schr\"{o}dinger-Poisson
systems with critical exponential growth. We establish conditions for the local
well-posedness of the Cauchy problem in the energy space, which seems
innovative as it was not discussed at all in any previous results. By
introducing some new ideas and relaxing some of the classical growth
assumptions on the nonlinearity, we show that such system has at least two
standing waves with prescribed mass, where one is a ground state standing waves
with positive energy, and the other one is a high-energy standing waves with
positive energy. In addition, with the help of the local well-posedness, we
show that the set of ground state standing waves is orbitally stable.
