My hunch is that the Romulan Neutral Zone and/or the Klingon Neutral Zone derive conceptually from those actual neutral zones.
I always took it the Neutral Zone was lived from the concept of the Demilitarized Zone between North and South Korea established in 1953 or the DMZ between North and South Vietnam established in 1954. Thanks for pointing out there were others.
In TOS the only Neutral Zone mentioned was the Romulan Neutral Zone. In Star Trek II the Neutral Zone suddenly became the Klingon Neutral Zone. This is because an early draft of the script had the Kobyashi Maru depicting Romulans until Harve Bennette changed them to Klingons. One evidence of this is that the Kobyashi Maru was stranded in the Gamma Hydra system. Gamma Hydra was depicted as belonging to the Romulans in TOS "The Deadly Years."
It is interesting to note that the Kobyashi Maru was, according to dialog, in the Neutral Zone.
In TOS "The Trouble with Tribbles," Sherman's planet was in disputed space. According to the Organian treaty, both the Federation and Klingons had equal right to see which society could best care for the planet. Also according to treaty the Klingons had full docking privileges at stations like K-7. No mention of a Neutral Zone was given in this episode.
Apparently it was only the Federation that couldn't violate the Neutral Zone. On screen evidence from TOS and TNG depicted the Romulans and Klingons violating the zone all the time. The Romulans and Klingons get trigger happy whenever a Starfleet vessel violates the Zone, but the Federation always wait until the enemy ships are on their side of the border.