If you offer information services in Germany you have to allow pseudonymous access if it is reasonable to do so, as part of privacy rules (principle of "Datensparsamkeit" = "data economicalness" - requiring as few pieces of information as absolutely necessary).
It is an interesting question if Facebook requires your real name to function properly and if this decision would hold in court, but it is in principle covered by law and I imagine the fact that facebook and other social networks have worked without these rules before might work against facebook here.
It is an interesting question if Facebook requires your real name to function properly and if this decision would hold in court, but it is in principle covered by law and I imagine the fact that facebook and other social networks have worked without these rules before might work against facebook here.