The bottle should reflect the name that's written by the MD on the script. Pharmacists are usually fairly stringent about having names and addresses of the person on the script.
It seems an odd thing, but I suppose mega-celebrities have to do extraordinary things for privacy.
A Hollyweird celebrity could to this and nobody would give it a second glance.
You're assuming Rush had the prescription filled by a pharmacist. More likely it was dispensed directly to him by his physician.