The Byzantines certainly practiced adoption! Orphans (or children whom widowed mothers could no longer support) could be cared for in monasteries or in actual orphanages (orphanotropheia) until they were old enough to marry, but adoption was also an apparently popular option. Children could be adopted to ensure the smooth continuation of a family line (particularly among the elite), due to general childlessness, or even simply philanthropy. Leo VI (r. 866-912) allowed even unusual groups - those that didn't or couldn't constitute a "traditional" married family unit, like eunuchs and single women - to adopt children (Novels 26 and 27), which seems to indicate that it was somehow a.) desired by these people and b.) something thought to benefit these people enough for the right to do so to be legally codified. As to whether this was a specifically noble practice, there are a number of famous elite adoptions across Byzantium - Zoe Porphrogenita's adoption of Michael V (1035), for example, or Michael Psellos' (1018-after 1081) adoption of a daughter - but I don't think it's unreasonable to assume this practice also occurred in the lower classes, too.

The Byzantines have been covered well. Adoption was not really a thing in Western Europe after the fall of the Roman Empire in the West. At least, not officially. Unfortunately we tend to know more about the nobility when it comes to such things. People in villages could have informally adopted orphans of local people all of the time without us knowing.

The English Common Law does not make clear adoption rules until the 1920s. Children were often "put out" in other homes where they had to work for the other family or learn their trade. If you are a young noble you may be put out in another house to be part of the staff. Where you would serve that Lord but also learn the customs and requirements of that position. This happened in noble families but not just noble families. And this could happen to orphans but not just orphans. It was one thing to be done with orphans. The Church took them in to be raised as lay people within monasteries. They could be put out or given over to apprenticeships (which is a little more formal than being put out) if they were an orphan of some means (but not nobility) or the Church or another institution was looking out for them.

If a child of the gentry was orphaned there would be a guardian appointed. The king (or queen) usually had the right to do this. And being the guardian for a well off orphan was coveted. This right was a big issue in the baronial revolt that led to the Magna Carta as King John tended to abuse it. It was so coveted because you could profit off of guardianship.

But if you were the guardian of a child then you didn't adopt them. There is no sense that you are now the son or daughter of this new family. You just looked over their affairs and made sure things were in order for them. They weren't your family now. At least not officially, I'm sure feelings of attachment developed sometimes. And they had no right to inherit from you.

I know England more than other countries in Western Europe on this issue. I know it was similar in countries like France, which at least beat Britain by having it first arise in the Napoleonic code. But even then it was very restrictive.

Like I said, this is all officially. I'm sure there were unofficial adoptions through out this time. Especially on the village level when families know each other for generations. But the official use of adoption in the West went out with Rome even though it continued in the Eastern Empire.

If you are a king and you want to have a general within your family you'd find some way to do a marriage alliance of some sort between your blood lines. Not adopt him.

For Early Medieval/Viking Period Northern Europe, adoption was indeed a thing because it served a political purpose. Royalty and freemen used it as a means of creating a trust between two households as it was considered a sacrifice for the one family to give up their child and an honor on the other part of the family to be allowed to raise their child. Of course some of these "adoptions" were giving up high profile collateral lest an agreement was broken. Another reason for adoption, at least in royalty, was to hide a child from political enemies that would want to kill the child give up their part in the succession line to be king or jarl.

One famous instance of adoption that I can think of is that of Haakon I "the Good" who given up by his father King Harald I "Fairhair" (the first unifying king of Norway) to King Athelstan of England. The reason for the adoption was King Harald wanted to hide his youngest son from his enemies but also from his own older sons who were vying for the throne. His one son Eirik "Blood-ax" was so called "Blood-ax" because it was believed that he killed most of his own kin to ensure that he would be granted the kingdom. While Eirik was crowned king after the death of Harald (which by some was believed to be killed by Eirik), Eirik was widely unpopular and when Haakon came back to Norway to claim the throne, Eirik was ousted and forced to flee to the Orkneys and then to Jorvik (modern day York, England) Haakon's adoption by Athelstan gave him the ability to learn the ways of being a good king from Athelstan which greatly helped his reign and also led him to being converted to Christianity. When Haakon came to the throne, he tried to Christianize Norway but was met with a lot of resistance.

This is post-Viking Era but Haakon IV (a lot of adopted Haakons) was raised and cared for under King Inge Bårdson. The story goes that Haakon was born to the birkebeiner (one faction in Norway during the civil wars) king Haakon Sverresson in bagler (the other faction) territory. As King Haakon Sverresson was dead at the time of his birth and bagler soldiers learned about the baby Haakon, they sought to go kill the baby to end the birkebeiner lineage. When birkebeiner supporters heard about this they had their best skiers Torstein Skevla and Skjervald Skrukka take the baby over a mountain from modern day Lillehammer to Østerdalen to give to King Inge Bårdson. This skiing escape is a popular story in Norway that is included in a painting and is also going to be in a new movie.