Traditionally we use a TRS to connect a headphone to an amp.
You can replace this by a TRRS connection so separate grounds for L&R
This is exactly how we connect our speakers to an amp and nobody calls this “balanced”.
This has of course nothing to do with “balanced” as we know it e.g. a 3 wire connection between and DAC and an amp where there is a hot, a cold and a ground.
At the receiver the hot and cold are compared and any difference is rejected (common noise)
The reason why in de headphone world a normal 2 wire connection is called “balanced” is probably because with a TRRS you can connect it to a balanced amp.
This is an amp requiring separate terminals for L&R
Using a "balanced" wire on a headphone won’t do you any harm, some even claim a small benefit (less crosstalk) but I haven’t seen any hard evidence.
In general the “balanced” out of a headphone amp doubles the power (2 amps per channel) but this might come at the expense of more distortion.
All the pro's of a balanced headphone amp:
https://www.headphones.com/pages/balanced-headphones-guideAll the cons:
https://benchmarkmedia.com/blogs/application_notes/audio-myth-balanced-headphone-outputs-are-better