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)
A Doctor Who fan once had a TARDIS mug which she used to drink tea out of. (WC: U.N.I.T. On Call)
In 4202, the Fifteenth Doctor brought two mugs with him to the Time Hotel. When hotel employee Trev Simpkins observed him with them, he asked if there was somebody else inside the TARDIS. The Doctor called it habit, referring to the fact that he did not have a companion at the moment, and handed one of the mugs to him, telling him to keep it, that it was bigger on the inside. (TV: Joy to the World [+]Steven Moffat, Doctor Who Christmas Special 2024 (BBC One and Disney+, 2024).)