This question calls to mind the real-life Australian case of Deing v Tarola in which the Supreme Court of Victoria "clarified the Weapons Act by stating that a studded belt is not a weapon when used for its intended purpose, but may become one if an offender intended to use it as a weapon."
The Improvised Weapon rules (SRD p65) state:
Sometimes characters don’t have their weapons and have to attack with
whatever is at hand.
This suggests that, like the studded belt in Deing v Tarola, an object becomes an improvised weapon when the character chooses to attack with it. If the character does not use the object as a weapon, then it is not a weapon.
A monk might therefore hold a pint of ale in one hand while using unarmed strike or a one-handed monk weapon.
Nevertheless, there ought to be some penalty for trying to fight while holding a pint of ale, at least if the character is attempting not to spill it. The Disadvantage rules (SRD p77) may apply here:
The GM can also decide that circumstances influence a roll in one
direction or the other and grant advantage or impose disadvantage as a
If it was my ruling to make, I would allow the Monk to use their Martial Arts ability, but with disadvantage on the attack roll.
(An exception might be made for a Drunken Master, but in that case a penalty should apply if they don't drink while fighting.)