Galaxy cluster number counts are an important probe to constrain cosmological parameters. One of the main ingredients of the analysis, along with accurate estimates of the clusters' masses, is the selection function, and in particular the completeness, associated to the cluster sample one is considering. Incorrectly characterising this function can lead to biases in the cosmological constraints. In this work, we want to study the completeness of the Planck cluster catalog, estimating the clusters' probability of detection in a realistic setting using hydrodynamical simulations. In particular, we probe the case in which the cluster model assumed in the detection method differs from the shape and profiles of true galaxy clusters. We create around 9000 images of the Sunyaev-Zel'dovich effect from galaxy clusters from the IllustrisTNG simulation, and use a Monte-Carlo injection method to estimate the completeness function. We study the impact of having different cluster pressure profiles, as well as that of complex cluster morphologies on the detection process. We find that the cluster profile has a significant effect on the completeness, with clusters with steeper profiles producing a higher completeness than ones with flatter profiles. We also show that cluster morphologies have small impact on the completeness, finding that elliptical clusters have slightly lower probability of detection with respect to spherically symmetric ones. Finally, we investigate the impact of a different completeness function on a cosmological analysis with cluster number counts, showing a shift in the constraints on $\Omega_m$ and $\sigma_8$ that lies in the same direction as the one driven by the mass bias.