Do I need a copper mug for a Moscow Mule?

No — a copper mug is the traditional vessel for marketing and presentation, but a highball glass or rocks glass works equally well for the drink itself. The copper is aesthetic, not chemical.

The copper Moscow Mule mug originated as a marketing tool in 1941: John Martin of Smirnoff Vodka and Jack Morgan of the Cock'n Bull restaurant had a friend with unsellable copper mugs, and the partnership took shape around using them for the drink. The copper does keep the drink very cold (high thermal conductivity), but only marginally faster than a glass; the real reason copper mugs persist is the visual — the matte orange contrast against the lime and mint is striking. Functionally, any tall glass works: highball, Collins, or rocks glass with extra ice. If using copper, never use unlined copper for acidic drinks — undiluted lime juice can pull trace amounts of copper into the drink. Most modern copper mugs are lined with stainless steel or nickel; check before buying. If the inside is shiny silver or matte stainless gray, it's lined and safe. If the inside is the same matte orange as the outside, it's unlined and should not be used for any cocktail with citrus.