When taking the ferry from Belfast to Liverpool do you need identification?
In general, yes you will need some form of identification, even if you are British or Irish (counterintuitively, as evidence of this fact, so that you then don't legally need identification), but it doesn't necessarily need to be photo ID such as a passport.
The precise rules will depend on your carrier, but to quote from the Stena Line FAQ:
I’M A BRITISH / IRISH CITIZEN. DO I NEED A PASSPORT TO TRAVEL?
British or Irish citizens travelling on our Irish Sea routes do not need a passport to travel to Britain or Ireland but are advised to take a form of identity. A driving licence, citizenship card or utility bill will usually suffice. A birth certificate will provide adequate proof of identity for your child to travel on our sailings.