Regular headphones with one plug have a contact point for L+, R+ and the ground/signal return shared between left and right. Balanced connections don't have the L and R sharing the signal return (and, as well, the signal return isn't shared with the ground, but that's another matter). This means only 4 pins/contact points are required: L+, L-, R+, R-. Since the original balanced headphone amp, made by Headroom, was two separate amps in one box, they used two plugs, choosing the regular 3-pin XLR connectors for it, but with the gender reversed. The third "ground" pin isn't used by headphones. The AKG K1000, however, used a single 4-pin XLR instead, though it was usually connected to the speaker terminals of an amp, but there was a special headphone amp made for them if I recall correctly, with a single 4-pin XLR socket. Since then, manufacturers and DIYers have used one or another system of 1x 4-pin or 2x 3-pin plugs, or both.
Balanced, in actual fact, has nothing to do with the headphones or plugs and really has to do with the topology of the amp driving them.