Given that neither Earth nor Romulus (supposedly) owns the entire Galaxy, such a claim can't be taken too literally. At best, the Zone extends until it reaches territory owned by third parties.
Alternately, the wording could quite literally mean that the RNZ is a sphere surrounding Romulus and Remus, separating them from the rest of the universe. Minimal "ownership" would then have universe-wide consequences.
The curvature shown in the "Balance of Terror" map would already be more indicative of a manageable eggshell RNZ than a grandiose flat-plane-that-divides-all-that-is one, IMHO.
Which could support an idea that there are limits to how far the Neutral Zone extends if those encounters take place in territory claimed by neither the Federation nor the Romulans.
There are political limits to the UFP's ability to enforce even a fully enclosing, eggshell RNZ. Namely, other empires border on the Romulans, too - Klingons are explicitly quoted as sharing a border, at which no "neutrality" is being practiced. All it would take to unleash the Romulans, then, would be another bordering empire that doesn't share even the tiny bit of understanding for the UFP that the Klingon Empire does...
It helps matters that Romulans found outside their RNZ confines tend to be violating interstellar laws, treaties or good manners in so many ways already that the actual Zone breach is seldom a prime concern!