Mage armor works with a shield. Shields are grouped with armor in the equipment rules in the Player’s Handbook, but various game features distinguish between the armor you wear and a shield you wield. Take a look at the monk’s Unarmored Defense feature and compare it to the barbarian’s version. In the monk’s version, you must both forgo wearing armor and forgo wielding a shield if you want to benefit from the feature, whereas a barbarian must only forgo wearing armor.