  We study the effect of self-interacting dark matter (SIDM) and baryons on the
shape of early-type galaxies (ETGs) and their dark matter haloes, comparing
them to the predictions of the cold dark matter (CDM) scenario. We use five
hydrodynamical zoom-in simulations of haloes hosting ETGs ($M_{\rm vir}\sim
10^{13}M_{\odot}$ and $M_{*}\sim10^{11}M_{\odot}$), simulated in CDM and a SIDM
model with constant cross-section of $\sigma_T/m_\chi = 1\ \mathrm{cm}^2
\mathrm{g}^{-1}$. We measure the three-dimensional and projected shapes of the
dark matter haloes and their baryonic content using the inertia tensor and
compare our measurements to the results of three $HST$ samples of gravitational
lenses and $Chandra$ and $XMM-Newton$ X-ray observations. We find that the
inclusion of baryons greatly reduces the differences between CDM and a SIDM,
together with the ability to draw constraints based on shapes. Lensing
measurements reject the predictions of CDM dark-matter-only simulations and
prefer one of the hydro scenarios. When we consider the total sample of lenses,
observational data prefer the CDM hydro scenario. The shapes of the X-ray
emitting gas are compatible with observational results in both hydro runs, with
CDM predicting higher elongations only in the very centre. Contrary to previous
claims at the scale of elliptical galaxies, we conclude that both CDM and our
SIDM model can still explain observed shapes once we include baryons in the
simulations. Our results demonstrate that this is essential to derive realistic
constraints and that new simulations are needed to confirm and extend our
findings.
