[Nyororin]

Another thing I forgot to mention.

As blood ties are important, it is not only rare for a parent to give up their child to a non-relative to be raised... It is also rare for there to not be a relative willing to take the child in... And even in cases where there is no family able to care for the child, it isn`t rare for there to be a family friend wanting to see the child raised by someone originally close to the family.
It is more common than many people realize for there to be siblings who are really "cousins". A lot of people assume the situation is something manufactured for anime - the common plot of a "brother" and "sister" living together who are either not related at all or very distant relatives falling in love, etc. The family situation is common enough that most people can think of someone whose family was that way.
(The falling in love part is entirely fabricated - the Westermarck effect pretty much rules it out unless there is something severely wrong with one of the siblings to begin with.)