A person who has a public office is called

A person who holds a public office is called either a civil servant or a politician.

Explanation:

A civil servant is someone who works for the Home Civil Service - a politically neutral organisation that advises and supports the government in delivering policies and public services.

A politician is a person active in party politics, or a person holding or seeking office in government ... Broadly speaking, a "politician" can be anyone who seeks to achieve political power in any bureaucratic institution.