I think you must be referring to 7.5.1.4 Preventive and Predictive Maintenance. It isn't specific how you have to set up your system but it gives minimum expectations for a maintenance system. An "asset register" which I guess I would normally call an equipment log is certainly a good start on tracking a maintenance schedule and history. There is no specific requirement for how you identify this equipment (asset number). You could probably get away with identifying a piece of equipment as "the big blue thing in the corner" as long as you all knew what you were referring to and kept all the required data. Tools under the maintenance program also require identification however there is no specific "numbering" requirement.
If you are responsible for maintenance on FLTs or injection molding equipment they must definitely be in your system (ID and all) if they are considered key process equipment (which I don't see how they wouldn't be unless they were sitting in the corner as a spare). Even if the leasing company did the maintenance, they should still be in your system for documentation of service, repair and other data.
I hope this answers your question. If not, please post again for clarification.
Dave