A shield requires an arm, not just a hand, to wield:
Anyone can…strap a shield to an arm.
PH, Armor and Shields, p. 144
This jibes with real-world use of the large shields, where the arm provides necessary support to make the shield useful. (Bucklers, or hand shields, may be a different issue, but there are no bucklers in official D&D 5e rules.)
“Manipulating an object” (Mage Hand, PH. p. 256) is different than wielding it.
Mage hand is not strong enough
The hand can only carry 10 pounds. A melee attack can involve thousands of pounds of force — orders of magnitude more than enough to overwhelm the carrying capacity of the hand. Even arrows project much more force than 10 pounds. (And since the hand is not connected to the ground, this force could bat the shield away like a piñata.)
This jibes with the spell description that says Mage Hand “can’t attack.”
If it can’t “wield” a shield, how about just holding it?
Just holding a shield up might convey some benefits to anyone behind it. Depending on the situation, the DM could rule that it provides concealment from view, or some other benefit.
But that’s not wielding a shield.