This paper seeks to advance linear stability analyses of thermoacoustic systems conducted with the stabilized Finite Element Method (sFEM). Specifically, this work analyzes and quantifies the impact of the Streamline-Upwind-Petrov-Galerkin (SUPG) artificial diffusion scheme on (eigen)mode shapes and damping rates of the isentropic Linearized Euler Equations (LEE) in frequency space. The LEE (eigen)mode shapes are separated into acoustic and vortical perturbation components via a Helmholtz decomposition and their sensitivity on the employed stabilization scheme is investigated separately. The regions where numerical stabilization mainly acts on the perturbation types are identified and explanations for the observations are provided. A methodology is established, which allows the quantification of the impact of artificial diffusion on the acoustic field in terms of a damping rate. This non-physical damping rate is used to determine the physically meaningful, acoustic LEE damping rate, which is corrected by the contribution of artificial diffusion. Hence, the presented method eliminates a main shortcoming of LEE eigen-frequency analyses with the sFEM and, as a result, provides more accurate information on the stability of thermoacoustic systems.