Galactic magnetic fields are thought to be seeded by microscopic processes such as the Biermann battery and are primarily amplified to observed microgauss strengths by dynamo processes. Much of this amplification likely occurs within proto-galaxies, where turbulent and rotational kinetic energy is converted into magnetic energy via dynamo action. However, two aspects of this process remain poorly constrained in simulations: the saturation level of the amplified field and the timescale on which it is reached. In particular, different numerical methods, resolution strategies, and subgrid prescriptions have not converged to similar results, making it difficult to establish robust, code-independent conclusions about dynamo amplification in galaxies. In this work, we present the first systematic comparison of both Eulerian and Lagrangian codes that employ diverse MHD schemes, including ART, RAMSES, AREPO, and SWIFT. Using identical initial conditions for Milky Way-type galaxies and without including star formation and stellar feedback, we find that the amplification history of the magnetic field within the galactic disk depends strongly on the choice of hydrodynamical scheme, with orders-of-magnitude differences between Eulerian and Lagrangian approaches. We further demonstrate that reducing the mesh velocity in the moving-mesh code AREPO leads to results consistent with those obtained using the static grid codes ART and RAMSES. These results highlight the importance of carefully assessing numerical effects when interpreting magnetic field amplification in galaxy simulations.