Is there any database system that allows to have different schemas for different users but the same content for all users?
Do you define "different content" as row filtering via security settings? That is pretty standard - and pretty unusual (as in: standard for high security, not all databases support this thinking)... SQL Server for example is just adding this with version 2016 due this year. In most cases... people do not bother as the filtering of what data is visible is done on the application layer (possibly in a server obviously).