There certainly is confusion between the lessons learned register and the lessons learned repository. Lets look at a PMP Exam Sample Question on this topic:
The register belongs to the project. It is an output of the process Manage Project Knowledge. As we collect lessons learned on the project they should be collected and put in the lessons learned register.
The repository belongs to the organization. It is an organizational process asset and an output of the process Close project or phase. At the end of each phase and or the end of the project the lessons learned in the register should be moved to the repository.
For the PMP® Exam it is important to know the difference between the lessons learned register and the lessons learned repository.