A City being Neutral is the rough equivalent of a Unit being a Barbarian. Neutral Cities and units within them are frozen in time. They do not progress production, they do not generate Shmuckers, and they do not belong to anyone. Once a unit enters the City Hex time resumes passing.
Because time resumes for all the Barbarians within the City it is unclear if approaching a Neutral Capital City could result in the Barbarians immediately claiming it and founding a new side, or if non-allied units present in the hex would prevent that.
Alternatively, it may not be appropriate to consider units within a Neutral city "Barbarian" rather than "Neutral", as Barbarians seem to have a wide latitude of possible actions that units in a Neutral city definitely lack. This may include the ability to claim a city and found a side, if only because they are unable to do anything at all if their zone of the city is uncontested (original source does not seem to confirm whether "can do nothing until attacked" applies to the whole city, particular zones, or maybe even individual stacks, no other sources seem applicable).
It also might be possible for Barbarians to enter and interact with a Neutral city (and possibly the remaining defending units) without attacking. This might allow Barbarians to use certain features of a city despite lacking the military strength to eliminate the remaining defending units without unacceptable losses. This may explain references to "Barbarian" cities despite the dissimilar meanings of Barbarian and Neutral. It would seem that such a situation would be quite temporary before a side became aware of the city and attacked and razed/claimed it.