The Kobayashi Maru describes its position to be ``Gamma Hydra, Section Ten'' ... with Gamma Hydra being right at the edge of Romulan space, per The Deadly Years. The preponderance of evidence is that the Kobayashi Maru scenario is set at the Romulan Neutral Zone.
The Kobayashi Maru does claim to be nineteen periods out of Altair Six, which per Amok Time had been an unsettled area of space. The presence of three starships there was meant to send a message to the Klingons. Therefore while the setting was the Romulan Neutral Zone, the presence of Klingons in space apparently close enough they might credibly contest it is surprising but not outrageous.
This is interesting. So when characters refer it simple "the Neutral Zone", it's consistent with the notion the Romunlan NZ being the only one NZ.
I'm inclined to conclude there is no Klingon NZ. Anyone have "Balance of terror" handy? any characters say "Romulan NZ" or simply "NZ"?