  We study $F$ coupled $q$-state Potts models in a two-dimensional square
lattice. The interaction between the different layers is attractive, to favour
a simultaneous alignment in all of them, and its strength is fixed. The nature
of the phase transition for zero field is numerically determined for $F=2,3$.
Using the Lee-Kosterlitz method, we find that it is continuous for $F=2$ and
$q=2$, whereas it is abrupt for higher values of $q$ and/or $F$. When a
continuous or a weakly first-order phase transition takes place, we also
analyze the properties of the geometrical clusters. This allows us to determine
the fractal dimension $D$ of the incipient infinite cluster and to examine the
finite-size scaling of the cluster number density via data collapse. A
mean-field approximation of the model, from which some general trends can be
determined, is presented too. Finally, since this lattice model has been
recently considered as a thermodynamic counterpart of the Axelrod model of
social dynamics, we discuss our results in connection with this one.
