  Models of two-dimensional (2D) traps, with the double-well structure in the
third direction, for Bose-Einstein condensate (BEC) are introduced, with
attractive or repulsive interactions between atoms. The models are based on
systems of linearly coupled 2D Gross-Pitaevskii equations (GPEs), where the
coupling accounts for tunneling between the wells. Each well carries an optical
lattice (OL) (stable 2D solitons cannot exist without OLs). The main subject of
the work is spontaneous symmetry breaking (SSB) in two-component 2D solitons
and localized vortices. We demonstrate that, in the system with attraction or
repulsion, SSB occurs in families of symmetric or antisymmetric solitons (or
vortices), respectively. The corresponding bifurcation destabilizes the
original solution branch and gives rise to a stable branch of asymmetric
solitons or vortices. In the model with attraction, all stable branches
eventually terminate due to the onset of collapse. Stable asymmetric solitons
in higher finite bandgaps, and vortices with a multiple topological charge are
found too. The models also give rise to first examples of \textit{embedded
solitons%} and \textit{embedded vortices}(the states located inside Bloch
bands) in two dimensions. In the linearly-coupled system with opposite signs of
the nonlinearity in the two cores, two distinct types of stable solitons and
vortices are found, dominated by either the self-attractive component or the
self-repulsive one. In the system with a mismatch between the two OLs, a
\textit{pseudo-bifurcation} is found: when the mismatch attains its largest
value ($\pi $), the bifurcation does not happen, as branches of different
solutions asymptotically approach each other but fail to merge.
