Feedback from active galactic nuclei (AGNs) has become established as a fundamental process in the evolution of the most massive galaxies. Its impact on Milky Way (MW)-mass systems, however, remains comparatively unexplored. In this work, we use the AURIGA simulations to probe the impact of AGN feedback on the dynamical and structural properties of galaxies, focusing on the bar, bulge, and disc. We analyse three galaxies - two strongly and one unbarred/weakly barred - using three setups: (i) the fiducial AURIGA model, which includes both radio and quasar mode feedback, (ii) a setup with no radio mode, and (iii) one with neither the radio nor the quasar mode. When removing the radio mode, gas in the circumgalactic medium cools more efficiently and subsequently settles in an extended disc, with little effect on the inner disc. Contrary to previous studies, we find that although the removal of the quasar mode results in more massive central components, these are in the form of compact discs, rather than spheroidal bulges. Therefore, galaxies without quasar mode feedback are more baryon-dominated and thus prone to forming stronger and shorter bars, which reveals an anticorrelation between the ejective nature of AGN feedback and bar strength. Hence, we report that the effect of AGN feedback (i.e. ejective or preventive) can significantly alter the dynamical properties of MW-like galaxies. Therefore, the observed dynamical and structural properties of MW-mass galaxies can be used as additional constraints for calibrating the efficiency of AGN feedback models.