  We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite
slope, defined as local pieces of an eigenvariety and equipped with a suitable
integral structure. Under some assumptions on the residual image, we show that
the image of the Galois representation associated with a family is big, in the
sense that a Lie algebra attached to it contains a congruence subalgebra of
non-zero level. We call Galois level of the family the largest such level. We
show that it is trivial when the residual representation has full image. When
the residual representation is a symmetric cube, the zero locus defined by the
Galois level of the family admits an automorphic description: it is the locus
of points that arise from overconvergent eigenforms for $\mathrm{GL}_2$, via a
$p$-adic Langlands lift attached to the symmetric cube representation. Our
proof goes via the comparison of the Galois level with a "fortuitous"
congruence ideal, that describes the zero- and one-dimensional subvarieties of
symmetric cube type appearing in the family. We show that some of the $p$-adic
lifts are interpolated by a morphism of rigid analytic spaces from an
eigencurve for $\mathrm{GL}_2$ to an eigenvariety for $\mathrm{GSp}_4$. The
remaining lifts appear as isolated points on the eigenvariety.
