Which mythological deity's 'brand identity' included a caduceus (staff with two intertwined snakes) that has been incorrectly adopted as a symbol for modern medical organizations?

Throughout history, mythological symbols have been powerful tools for identity and recognition—much like modern brand logos and color schemes. Ancient gods, heroes, and cultures used distinct visual elements that functioned similarly to today's brand identities. This trivia explores how mythological figures established their 'brand identity' through specific symbols, colors, and attributes that made them instantly recognizable across different cultures and time periods.

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Hermes/Mercury (messenger god of commerce and travelers)
Apollo (god of healing and medicine)
Asclepius (god of medicine and healing)
Athena/Minerva (goddess of wisdom and strategic warfare)