Regent Street is one of several names which may have been transferred from London to Belfast, such as Oxford Street, Bond Street and Fleet Street (the last two no longer extant). Each possible transfer needs to be evaluated individually. Regent Street in London was named for George, Prince of Wales (later King George IV) and laid out by the architect John Nash between 1813 and 1821 as part of a scheme for the prince (Room 1994, 451). The fact that Belfast’s Regent Street was named in 1863, four decades later during the reign of Queen Victoria (when there was no regent), speaks for this being a transferred name. See this blog: Transferred Street Names In Belfast
The Irish version of this name has been revised. It was rendered Sráid na Ríóna when a dual language form was first approved.