This alone may not be sufficiently proven by an ID document like driving license, and some airlines will only accept passport for travel.
https://help.ryanair.com/hc/en-gb/articles/12889174472721-Wh...
Having a long term residency visa in the UK, but an Indian passport for example, would not automatically qualify you for travel to Ireland without a separate visa, although lax checks within the CTA may, in theory, let you travel if no one looks hard enough and should an airline be satisfied with your UK driving licence alone (and assume you're either an Irish or British citizen).
The requirment is further complicated by the fact that some third country nationals holding either a used and valid UK or Irish short term (tourist) visa can travel to either countries without issues due to the British-Irish Visa Scheme/treaty, but the same courtesy wouldn't apply to a third country national with a (long term) UK BRP - they'd need a separate visa for Ireland.
Border checks or lack thereof in the CTA are a rather interesting subject matter.