The Staff of Defense (LMoP) says:
The staff has 10 charges, which are used to fuel the spells within it. With the staff in hand, you can use your action to cast one of the following spells from the staff if the spell is on your class’s spell list: mage armor (1 charge)...
Since your wizard attuned to the staff, and mage armor is on the wizard spell list, the wizard may use the staff to cast mage armor.
The DMG section on "Activating a Magic Item" (p. 141) says:
SPELLS. Some magic items allow the user to cast a spell from the item. The spell is cast at the lowest possible spell level, doesn't expend any of the user's spell slots, and requires no components, unless the item's description says otherwise. The spell uses its normal casting time, range, and duration, and the user of the item must concentrate if the spell requires concentration.
Because the description of the staff says that the user may cast the spell from it, fueled by charges, no spell slots are used for this casting (and the wizard does not have to have the spell prepared or even known).
The mage armor spell has range of "Touch" (not "Self"), so the wizard can use the staff to cast the spell on themselves or another creature.
Thus, the only things limiting the number of people on whom the wizard may have the spell running are the charges in the staff, and as Erik notes, the duration of the spell itself.