  It is well known that the two-dimensional (2D) nonlinear Schr\"odinger
equation (NLSE) with the cubic-quintic (CQ) nonlinearity supports a family of
stable fundamental solitons, as well as solitary vortices (alias vortex rings),
which are stable for sufficiently large values of the norm. We study stationary
localized modes in a symmetric linearly coupled system of two such equations,
focusing on asymmetric states. The model may describe "optical bullets" in
dual-core nonlinear optical waveguides (including spatiotemporal vortices that
were not discussed before), or a Bose-Einstein condensate (BEC) loaded into a
"dual-pancake" trap. Each family of solutions in the single-component model has
two different counterparts in the coupled system, one symmetric and one
asymmetric. Similarly to the earlier studied coupled 1D system with the CQ
nonlinearity, the present model features bifurcation loops, for fundamental and
vortex solitons alike: with the increase of the total energy (norm), the
symmetric solitons become unstable at a point of the direct bifurcation, which
is followed, at larger values of the energy, by the reverse bifurcation
restabilizing the symmetric solitons. However, on the contrary to the 1D
system, the system may demonstrate a double bistability for the fundamental
solitons. The stability of the solitons is investigated via the computation of
instability growth rates for small perturbations. Vortex rings, which we study
for two values of the "spin", s = 1 and 2, may be subject to the azimuthal
instability, like in the single-component model. We also develop a
quasi-analytical approach to the description of the bifurcations diagrams,
based on the variational approximation. Splitting of asymmetric vortices,
induced by the azimuthal instability, is studied by means of direct
simulations. Interactions between initially quiescent solitons of different
types are studied too.
