Nice to have my lame neophyte guesses on translation confirmed by someone with more clue base.
As for what that *means*, personally, I've always interpreted Mai's bond to Sayuri as a really viscerally emotional soul-bond (she really couldn't carry on without Sayuri) - to use the greek -- "agape" rather than "eros". Obviously fan fiction can go nuts with it, but I think they're interpreting incorrectly
Anyone who chooses Mai is, by default, going to have to deal with Sayuri being inextricably in the picture.... not such a bad deal really.