Wrapper classes use composition. The same composition as in the popular maxim, "Favor composition over inheritance." Composition is not a design pattern; however, most OO design patterns use composition as part of their implementation. This is one reason many people struggle to discriminate among different design patterns: the common use of composition makes them all appear the same to a certain degree.
There are two basic parts in a composition relationship: the composer and the composed. You can think of this generally as a part/whole relationship. It may be one-to-one or one-to-many. A wrapper is the composer, i.e. it is the whole. It may wrap one or more composed parts.
Many different design patterns make use of the general composition relationship for different purposes. Many of those different patterns are referred to collectively as "wrappers". The GoF book calls out at least two such patterns.
ADAPTER Also Known As Wrapper
page 139
DECORATOR Also Known As Wrapper
page 175
In summary, Wrapper is not any single design pattern; rather, it is a category of design patterns. Incidentally, we see the same dynamic with the term Factory. There is no single design pattern named Factory; rather, it is a category of design patterns.