Hi Tzu
The excellent Copper Family are probably unique in Britain. Their style seems to have evolved in some way from the glee clubs of the early nineteenth century although I strongly suspect developed further by the family. I say this because some of their songs are indeed from this glee club repertoire but most are certainly not. There are plenty of families who have been recorded right up to the present day, but all the ones I have heard of have been solo singers with the occasional solo singer being accompanied on instrument by another member, but even this is rare. Hundreds of singers have been recorded either in writing or by sound recording since the 1880s and to the best of my knowledge there are no others.

However, arguably there have been performing families of musicians in Ireland who have sung together, such as the McPeakes. Whether they sang together in this way before they became performers I couldn't say.