It seems "brainchild" is consisted of "brain" and "child". Is this word creatively made by speaker?
No, "brainchild" is a commonly used informal noun. It means the idea or invention known to be a specific individual's work or creation. For example, SpaceX is the creation of Elon Musk, so it is considered his "brainchild." Ex: The new startup was the brainchild of a child genius. Ex: The proposal is the brainchild of my supervisor.