Families would adopt frequently to maintain the family line, and these adoptions would typically involve adopting a relative from a greater family tree rather than a true "unknown." One should note that just because the term is adoption, it does not mean that these children were adopted as babies as one might assume (as is common today in the West.) The term "adoption" might be used in English to refer to a variety of processes in China, from "adoptions" for the purposes of alliance-building, to adoptions for the continuation of the family line, and also to a variety of levels of formality. In some cases, adoptions would include the taking of a new clan name. In other cases, the "adoptee" would maintain their own family name. In addition, adoption wasn't necessarily a permanent affair.

I haven't watched the drama you're talking about so I don't know how they portray these relationships, but it's quite possible that many relationships that would not be called "adoptions" in the Western sense are also called "adoptions" because they are a type of adoption in the Han sense. That could contribute to why you think there's such a frequency of adoptions in the Three Kingdoms Era/ Han Dynasty. Many of these "adopted children" probably had families and weren't adopted as babies.

Edit: I gave a very brief answer (for time constraints) but if you're interested in details, depending on what you're interested in I can certainly try my best to provide more info!