Yes, uniquely identifying the patient is important, especially for pharmacies where people with the same name might receive different doses of the same drug or receive similar sounding drugs that the patient might not catch.

What would you suggest? It needs to be a piece of information that is probably unique when paired with name at least as far as the local area, that absolutely everyone has, that the pharmacy and doctor both have and is unlikely to change (to avoid issues where records in one place are updated before the other).