What countries allow adult adoption?
As far as I know, adult adoption is legal in the U.S., but the courts would look at the particular situation to ensure that it was not being used to exploit vulnerable people or circumvent laws related to marriage and sexual relationships.
The main use of adult adoption in the U.S. is to provide for the ongoing care of an adult who, by virtue of a severe physical, cognitive, or emotional/behavioral special need, will never be able to function independently.
As an example, a married couple gives birth to a son who has serious cognitive and developmental limitations. Despite the negative comments from some relatives and acquaintances, who think that the child should be institutionalized, the husband and wife devote themselves to making the most of their son's capabilities and helping him lead a happy life.
The child becomes a man in chronological terms, but he remains a toddler in many ways. He is nonverbal, needs assistance with walking, uses diapers. He receives a variety of support services in the home, and participates in a special school program for the profoundly disabled, where he enjoys music, art, physical therapy, and other activities.
Eventually, the man turns 40. His parents are now 70, and have some health issues. The husband has had a heart attack, and is no longer able to help lift and support his son easily. The wife has chronic bronchitis. They are forced to face the reality that they are likely to die while their son remains alive, and worry that he will be placed in an institution, where he almost certainly will not do well .
Fortunately, through their church, they recently met a young couple in their early 30s. Both spouses grew up with siblings who had severe brain damage related to hypoxia during delivery, and who died in their teens. Now working for disability related organizations, they decided that they didn't want to have biological children; they wanted to adopt either an older child or an adult who had cognitive impairment or severe brain injury, and who otherwise would wind up in an institution.
The young couple recently met the older couple's son and felt very strongly that they would enjoy having him as their son. The two couples are now working with an attorney on arrangements that would gradually transition the older couple's son to the younger couple's care, and that would culminate in a legal adoption of the son by the younger couple.
Now there are other situations, where adult adoption sometimes occurs in the U.S, but they are less common. As an example, suppose a man married a widow who had three children. Two of the children were minors at the time; the third had recently turned 21. The man agreed to adopt the minor children. Technically, he did not have to adopt the oldest child, who was already an adult and self-supporting, living in another state. However, the oldest child asked if he could be adopted too, because he wanted to feel that same connection with his new stepfather that his siblings had. Realizing how important this was to the oldest son, the mother and stepfather readily agreed, and the judge included the oldest son in the adoption decree.