How many things can a person hold and use at one time?

I’m learning to play / DM, and watching various videos on line and experimenting. One scenario came up on which I’d appreciate clarification:

The situation is a human wizard has descended some stairs into a dark room. They have taken a torch from the wall in the room above, and they’re carrying their staff in the other hand. They get down to the lower room, and find a handaxe on the floor. The character says: "I pick up the hand axe". My question is: does that imply that the player is now holding two things in one hand? I can’t imagine somehow stowing the staff in my robe… nor can I hold the torch in my teeth — sure I could probably manage to hold both a torch and an axe, or some other combination of two things at once, but I doubt I’d be very proficient… maybe the character switches the staff to the hand with the torch, and uses their dominant hand to use the axe?

How would you handle this as a DM? Just hand-wave it? Ask where the player is going to put either the torch or the staff? Or call out that they’re carrying the torch and the staff in one hand.. make a dexterity check to see if the torch is dropped? Or am I just over-thinking this?

As the action unfolds, the wizard then casts a spell with a somatic component, despite having three things in two hands… they then throw the axe (now just maybe they have two things in one hand) and cast another spell with a somatic component…

Am I just being pedantic here?