UID in DICOM (no matter what UID) is always globally unique. So, as you asked in question, uniqueness is not limited to Patient level or something.
Following is from specifications:
2017a Part 5 - Data Structures and Encoding (9 Unique Identifiers (UIDs))
Unique Identifiers (UIDs) provide the capability to uniquely identify a wide variety of items. They guarantee uniqueness across multiple countries, sites, vendors and equipment. Different classes of objects, instance of objects and information entities can be distinguished from one another across the DICOM universe of discourse irrespective of any semantic context.
More details about DICOM UID can be found in this answer.
Your comment on question as below:
My question was more about what to do in case I choose to clone a patient in my system and attach the same dicom(s) to it. Should I regenerate the dicom-uid's or could I keep them as-is.
I am not sure what you mean by "clone". While cloning, if there is change in dataset, you should regenerate the SOPInstance UID. Even if you simply apply lossy transfer syntax to your dataset, you should regenerate the SOPInstance UID. Any action that differentiates/separates the the datasets from original require new SOPInstance UID. So, while cloning, if you are changing patient demographics, new UID should be generated. Whether new StudyInstance UID should be generated or not depends upon what is changed.
OTOH, if you are just copying your dataset at different location, it is still same dataset. You do not need to regenerate UIDs in this case.