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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 $σ_T/m_χ= 1\ \mathrm{cm}^2 \mathrm{g}^{-1}$. 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.