The Republic of Northern Cyprus is a separate state with its borders, parliament, laws and president. The state is based on British law and all foreign nationals (Turks among them) across the country’s border on the same grounds.
Therefore, the prefix to the name Turkish does not refer to the Republic as a Turkey part. Historically, the island Turkish population (Turkish Cypriots) has expressed a desire to establish a state.
Back in the 1960s, ethnic conflicts between the island Turkish and Greek populations began, which ended in a military coup. Greece, Turkey, Great Britain and the UN actively took part in the confrontation. When a referendum was held in 1974 on the Northern and South Cyprus federal-state creation, the population majority (Greek Cypriots) did not support the idea, in 1983 the Northern Cyprus empire was formed but is now recognized only by Turkey.
Thus, the Turkish Republic of Northern Cyprus does not belong to Turkey but is an independent country formed at the request of the island Turkish population. With its economy, language, capital, state government and legal system.