(Abridged) Context. Most massive stars are located in multiple stellar systems. Magnetic fields are believed to be essential in the accretion and ejection processes around single massive protostars. Aims. Our aim is to unveil the influence of magnetic fields in the formation of multiple massive stars, in particular on the fragmentation modes and properties of the multiple protostellar system. Methods. Using RAMSES, we follow the collapse of a massive pre-stellar core with (non-ideal) radiation-(magneto-)hydrodynamics. We choose a setup which promotes multiple stellar system formation. Results. In the purely hydrodynamical models, we always obtain (at least) binary systems. When more than two stars are present, their gravitational interaction triggers mergers until there are two stars left. The following gas accretion increases their orbital separation and hierarchical fragmentation occurs so that both stars host a comparable disk and stellar system which then form similar disks as well. We identify several modes of fragmentation: Toomre-unstable disk fragmentation, arm-arm collision and arm-filament collision. Disks grow in size until they fragment and become truncated as the newly-formed companion gains mass. When including magnetic fields, the picture evolves: the primary disk produces less fragments, arm-filament collision is absent. Magnetic fields reduce the initial orbital separation but do not affect its further evolution, which is mainly driven by gas accretion. With magnetic fields, the growth of individual disks is regulated even in the absence of fragmentation or truncation. Conclusions. Hierarchical fragmentation is seen in unmagnetized and magnetized models. Magnetic fields, including non-ideal effects, are important because they remove certain fragmentation modes and limit the growth of disks, which is otherwise only limited through fragmentation.