Kaouru should know if such possibility existed, he's not the first generation having such abilities
My theory is that people with the ability never stay too long near their (twin) siblings to find out that they actually do have the string. Kaoru himself resents his brother to some extend, since he found his soulmate and built a family, while Kaoru is stuck losing everyone he falls in love with. It's never mentioned about their family, so that's what I think: before the new generation develops the power, the older one kinda just loses touch, so they never realized it.
People have been pointing out that Kei couldn't see the string in Kaoru's finger, which means he doesn't have one. Well, I would like to present a new theory: Kaoru doesn't have a string for as long as he is the one with the power; however, as soon as one of the twins develops it, Kaoru's string will "be born" or, more precisely, it will become visible. Maybe Kaoru even loses his ability of seeing strings once a new person acquires the ability, but you get my thing: Kaoru and Hara are soulmates in the end and he will find out when the twin points out Kaoru's string attached to someone already ヾ(☆▽☆)