What is Fosterage?

Legal Definition
Fosterage, the practice of a family bringing up a child not their own, differs from adoption in that the child's parents, not the foster-parents, remain the acknowledged parents. In many modern western societies foster care can be organised by the state to care for children with troubled family backgrounds, usually on a temporary basis. In many pre-modern societies fosterage was a form of patronage, whereby influential families cemented political relationships by bringing up each other's children, similar to arranged marriages, also based on dynastic or alliance calculations.

This practice was once common in Ireland, Wales, and Scotland.
-- Wikipedia
Legal Definition
Rearing a child not one’s own as one’s own.
-- Ballentine's Law Dictionary