Are PhD students enrolled or hired? In most universities, PhD students are literally students similar to master students. Fellowships and scholarships only support them financially, but officially they are students.

Some European university advertise PhD open positions like a contract job (similar to research associates or postdoc). It is understandable that these funded PhD students provide some services for the university, but in the first case too, PhD students regularly provide TA and RA services, but they are still students.

Are these two schemes different in action, or difference is only in words on paper?

Do European PhD students (mentioned in the second case) have more rights or flexibility?
At least in Germany, doctoral candidates are either employed as Wissenschaftliche Mitarbeiter (research assistants) or given stipends as "doctoral fellows."

However, regardless of how they are paid for their efforts, they must "enroll" as doctoral candidates with one of the faculties if they are to receive a degree. So, to a certain extent, they are treated like students as well, and have many of the same privileges (for instance, they are entitled to the same student benefits when they register).

So, to a certain extent, they are both; however, their employment contract, rather than student regulations, tends to be the "controlling factor" in their treatment.