How can a horse move a cart if they exert equal and opposite forces on each other according to Newton's third law?
You are overlooking the force between the horse and the ground.
Yes, if you had only the horse and the cart with nothing else, then the system could not accelerate as a whole (i.e., the center of mass could not accelerate). However, friction between the horse and the ground pushes on the horse, thus allowing for an acceleration of the system.
Since the two objects are attached together, they are technically the same object, and they cannot accelerate.
That's not really how it works here. That's like saying since a car is the same object, it cannot accelerate itself relative to the ground.
If the horse were rigidly attached to the cart (as opposed to "tethered", which implies a rope), it would struggle to accelerate relative to the cart, but both the cart and the horse can be accelerated relative to the ground by applying force on the ground (when the horse runs).
You forget that horse also applies force on the ground so that it can move together with cart. The forces between horse and cart only keep them relatively steady with respect to each other.