I can think of one instance that there is a description of this practice, the character of Aslaug in the saga of Ragnar Lothbrok. She was, according to the story, raised by her abductors who murdered her foster father. She lived with them until she was discovered by Ragnar's henchmen, they told him about her and she eventually wound up as his queen. Now her story appears in the 13th Century saga of Ragnar and was, well, a story. The events of her fictional life do not necessarily reflect reality in the practices of the Viking Age Norse.

If you are curious about whether the Norse people would abduct children on their raids and raise them as their own children, the answer is rather clear. No. The purpose of taking captives abroad for Norse raiders/Vikings/Pirates whatever you want to call them was rather clear, and it was for the slave trade or to be ransomed.

The first case is relatively clear cut. The Norse would abduct individuals, especially Church figures to extort a ransom from the Church/State that they had raided. These men, assuming they were monastic brothers or some other figure in the Church, were essentially moveable wealth, and were going to provide a return on the investment (of raiding). The Church in England especially was quite able to engage in ransom raising efforts to return captured clerics home. It is likely that among the men who were kidnapped by the Norse there were also novitiates and other young men/boys who were likewise taken by the Norse as a part of their ransom efforts.

The other major fate for those abducted by the Norse was the slave market. The Norse involvement in the slave trade was widespread, from Ireland to Baghdad, and the Norse provided humans for the slaves trade of the early Medieval world. Ireland and Russia in particular were centers for slaving and they were sold in the markets of cities like Bristol, Dublin, and Baghdad. From these centers the humans who were forcibly taken from their homes were bought and sold like other goods and used for domestic work, agricultural work, and sexual exploitation.