Probably from Scandinavian mugg 'jug' — the 'face' slang came from grotesque faces painted on drinking mugs.
A large cup with a handle, typically used for hot drinks.
Probably from a Scandinavian source, compare Swedish mugg 'mug, jug.' The 'face' sense (ugly mug) appeared around 1708, possibly from grotesque faces painted on drinking mugs. The 'to mug' (rob) sense came later still, around 1818. Key roots: mugg (Scandinavian: "mug, jug").