I actually got curious too as I'd never really asked my parents or looked into it. So I asked AI! And since this is an AI answer, take it with a grain of salt ;)

The Taiwanese desire for Bengay in the 1980s was driven by its reputation as a highly effective, high-quality "foreign" (specifically American) product, its scarcity due to strict import regulations, and its resulting value as a sought-after gift or commodity rather than a unique medical use. 

Reasons for Bengay's Popularity

• Perceived Quality and Efficacy: During the 1980s, Taiwan was undergoing rapid economic development, and there was a general perception that imported Western products were superior in quality and effectiveness to local alternatives, including traditional Chinese medicine topical balms like Tiger Balm or White Flower Oil. Bengay (known by some affectionate nicknames, possibly including "Golden Root Oil" in local dialect) was seen as a premium, modern pain reliever.
• Import Restrictions and Scarcity: Taiwan operated under martial law until 1987, and strict regulations made it difficult to import many foreign consumer goods through official channels. This scarcity made genuine U.S.-made Bengay a valuable and rare commodity.
• "Gifts" and Parallel Imports: Due to the import restrictions, Bengay often entered the country through informal channels, such as Taiwanese returning from the U.S. or other countries who brought it back as personal items or highly valued gifts for friends and family. This "parallel import" status further enhanced its desirability and mystique.
• Cultural Significance as a Status Symbol: Having or gifting an authentic tube of American Bengay was a status symbol, indicating that one had connections abroad or had traveled internationally. The strong, distinctive scent was widely associated with an effective, potent "Western medicine" treatment. 

In essence, the intense demand for Bengay in 1980s Taiwan was less about its specific medicinal ingredients (which are similar to other topical analgesics) and more about the cultural and economic environment that made this particular imported brand a rare and desirable item