A mug was a container for liquid.
The Third Doctor once made a time flow analogue from a Moroccan burgundy bottle, spoons, forks, corks, key rings, tea leaves and a mug. (TV: The Time Monster)
According to a commemorative mug that barmaids Susan and Judith once showed the Fifth Doctor, 24 June 1483 was the day when Edward V of England was to have been crowned. (AUDIO: The Kingmaker)
The Tenth Doctor once used a pencil to demonstrate how the Huon particles in the heart of the TARDIS were attracted to the identical particles in Donna Noble, using a mug to represent the TARDIS. (TV: The Runaway Bride)
Jason Kane was served with a mug that was a faithful representation of Marvin the Martian with a handle on it. (PROSE: Oblivion)
Whilst visiting the West Country Children's Home in the 1990s, the Twelfth Doctor took an employee's coffee mug while they weren't looking. (TV: Listen)
The Thirteenth Doctor had a mug that said "Universes #1 Dad" on it. (COMIC: A New Beginning)