Derived from the word polis ("city-state"), it has a range of meanings from "the rights of citizens" to a "form of government".
According to Liddell and Scott's Greek-English Lexicon a meaning of politeia is "the conditions and rights of the citizen, or citizenship", analogous to the Latin civitas.
Women, slaves and others who Greek men deemed unworthy were not in the active matrix of the political formations of that city state, making them not-citizens, so not part of politeia.
In the works of ancient Greek philosophers, the principal meaning of politeia appears to be: "how a polis is run; constitution".
Specific translations of this second meaning as constitutional democracy or republic are at least anachronistic, and in most instances contentious and/or inaccurate.
Note that the meanings the ancient Romans attached to res publica were also multiple and only partially overlapping with the Greek politeia, and further that few of the multiple meanings of politeia or res publica are much of an equivalent to republic as it is understood in modern political science.
Paul's father might have obtained "citizenship" (1) by manumission; (2) as a reward of merit; (3) by purchase; the contrast implied in Acts 22:28 is perhaps against the last mentioned; (b) "a civil polity, the condition of a state, a commonwealth", said of Israel, Ephesians 2:12.