The
记 character at the end of brand names (
徐福记,
鸿茅记...) is a traditional 'shop / make of...' suffix — basically 'Xu Fu's Brand'.
徐福 is a stock proper-name from Chinese history (a Qin-dynasty alchemist sent by
秦始皇 to find the elixir of immortality), but the brand is just borrowing its auspicious sound.