A child who is more connected to a father or king that rules eally well will be more suitable regardless of adoption or not. If the kingdom likes the king/father they will likely prefer someone like him or someone who will rule like him (or better) as replacement.

Children are usually more connected to their parents because they are designed in similar manner, have more compatible behaviors(such as likes and dislikes) programmed/trained by the parents in ways they preferred. If the adopted have these characteristics more than the blood children they would likely be preferred by the king or father. Ofcourse, it's possible for a father to adopt someone because he is like him, or they are very compatible such that they both like or dislike what each other like or dislike.
By the way, what counts most is the connection with the kingdom's cherished values. If the father does well promoting the values and making the kingdom better, people would be ok to see a replacement like him regardless where he comes from.

It will quite hard to find adopted with these characteristics, but it's possible, especially in Christianity. An adopted son who loves the CREATOR of the Universe more than his other siblings could be made the Priest or Leader of both his adopted family and Christian community.