That the author used two batteries in series did not change much in that respect (and internally those 9V batteries are a stack of 6 cells anyway). The alternative would be to use the common rail between the two batteries as a ground and then two regulators (or simply a Zener and a follower transistor, same effect for less cost) to get say plus and minus 7 or 8 volts depending on how far you want to run the batteries down and what kind of specs that op-amp has.

Most op-amp circuitry would be +- 15V, but they usually work on much less than that. The circuit as presented here probably has a DC bias that is in the 100's of mV, there is no adjustment for it either so it is a bit of a kludge the way it is set up. There are better ways to do this, it all comes down to part variation and that should not be a factor in a good schematic.

There are single supply op-amps too, but in this case I suspect the voltage required to drive those headphones properly was the major deciding factor in using two stacked 9V cells.