They were contemporary court writers from Japan's Heian period. In-game, Murasaki doesn't like Sei, but Sei thinks Murasaki's her best friend. Fandom likes matching them up because they're complete opposites personality-wise.
Sei and Murasaki serve in the same court at almost the same time, with Sei being the senpai that retired before Murasaki starting. But they serve a different ladies, so there's some sense of rivalry going on between them. Murasaki seems to really admired Sei, however (at least in her writing skills), as Murasaki used Sei's invented phrase "haru wa akebono" many times in her own writings.
