Black Lace were a 1980s band who sang party songs that were often derided for being naff, cheesy and irritatingly catchy (eg Agadoo). Although it seems unlikely that someone as sophisticated as Sarah Barnaby would have owned any of their albums, she would have been the right age as a girl to have known the songs and their dance moves. And someone like the more Eeyore-ish Barnaby would definitely roll their eyes at anyone who liked such upbeat, holiday songs.
In other words- the joke is based on culture clash.