@GingaBishoujo @wolfeng It was indeed her cousin she got together with. "Just after
someone else confessed to me" suggests that someone had indeed confessed to her, but it was after that when her cousin came and asked to be with her always. It's likely that either she rejected the first confession right before she got together with her cousin, or it's because they got together that she rejected the previous confession.
As for their incestuous relationship, relationships between cousins isn't as uncommon or as bad as one may think, although there's still a clear issue with inbreeding and genetic problems their children could've ended up with had Lily given birth before she died. With that said, all she mentioned was that they were cousins, but it wasn't specified to what extent. If her family was particularly large, then they could have easily been second or third cousins, where they were related through their grandparents or great-grandparents. The further the relation, the less of a problem it is regarding both the inbreeding and the societal taboo. Most importantly, however, is what kind of society they lived in. Some societies/cultures outright murder people who are known to be in an incestuous relationship, whereas others may tolerate it or, in more extreme cases, outright encourage it. At least, I believe in the distant past it was encouraged by certain cultures and/or religions, but I'm not so sure if there are modern day cultures/religions that do that still.
Despite everything I've said, please note I'm not an expert on this subject, so take it all with a grain of salt. I'm just sharing my understanding based on what I've read and learned.