  We study non-planar corrections in two special $\mathcal N=2$ superconformal
$SU(N)$ gauge theories that are planar-equivalent to $\mathcal N=4$ SYM theory:
two-nodes quiver model with equal couplings and $\mathcal N=2$ vector multiplet
coupled to two hypermultiplets in rank-2 symmetric and antisymmetric
representations. We focus on two observables in these theories that admit
representation in terms of localization matrix model: free energy on 4-sphere
and the expectation value of half-BPS circular Wilson loop. We extend the
methods developed in arXiv:2207.11475 to derive a systematical expansion of
non-planar corrections to these observables at strong 't Hooft coupling
constant $\lambda$. We show that the leading non planar corrections are given
by a power series in $\lambda^{3/2}/N^2$ with rational coefficients. Sending
$N$ and the coupling constant $\lambda$ to infinity with $\lambda^{3/2}/N^2$
kept fixed corresponds to the familiar double scaling limit in matrix models.
We find that in this limit the observables in the two models are related in a
remarkably simple way: the free energies differ by the factor of $2$, whereas
the Wilson loop expectation values coincide. Surprisingly, these relations hold
only at strong coupling, they are not valid in the weak coupling regime. We
also discuss a dual string theory interpretation of the leading corrections to
the free energy in the double scaling limit suggesting their relation to
curvature corrections in type IIB string effective action.
