This is a from a HSK6 past exam (paper H61005, page 18):
每一个中华老字号都是一个品牌奇迹，“同仁堂”有 300 多年的历史，京城最老的老字号“鹤年堂”的历史超过 600 年。今天它们不仅仅是品牌，更是一种文化，当我们提起这些品牌，最津津乐道的还是那些被人熟知的动人故事。
这就是家喻户晓的中华老字号品牌——同仁堂诞生的故事。1669 年，同仁堂药铺落成。时过 341 年，这间百年老店深深影响着几个世纪的中国人，首先影响到的便是听到同仁堂诞生故事的人们，一家平民小店与皇室的故事，一个小郎中打败宫中御医的故事，故事中的高超医术夹杂着传奇色彩，看病的或听故事的人纷至沓来。 ...
It tells a story of Emperor 康熙 getting sick, and only a doctor working at a 小药铺 (~small pharmacy stand) could cure him. He rewards the doctor by giving him a magnificent pharmacy, calling it 同仁堂.
About this snippet it asks the following question (with the given answer in bold):
This question confuses me for two reasons:
诞生 literally means "to be born", but is often used metaphorically to mean something like "to be brought into existence" (as per my understanding). So it appears the question asks about what 同仁堂 was before 同仁堂 existed, which is nonsensical---by definition, it didn't exist! So I feel like the answer could be 同仁堂诞生之前是不存在的。
The story says that the pharmacist's 小药铺 existed before 同仁堂, thus 小药铺 and 同仁堂 are two distinct entities (otherwise 康熙 couldn't give it to the pharmacist, since it already belongs to the pharmacist). Analogously, if I own an housecat, and someone gives me a tiger, it's obviously incorrect to say "before my tiger existed, it was a housecat"---they're two different things.
Maybe I'm misunderstanding something here.
Question: How can I understand the seemingly illogical HSK6-exam answer: 同仁堂诞生之前是：一个小药铺?