There isn't one.
In the technical context of the OP's full question there is no accepted single word technical term.
Many of the words mentioned are great for general responses, especially in terms of the question title, but would be questionable usage in a more specific technical context.
If this is referring to a tree graph structure specifically, it might be indicated by referring to the node in question being capable of having a
branch factor of two or higher (slightly awkward, as branch factor is more commonly used to describe a currently existing node or tree, not its potential), or having unlimited branching.
In terms of object descriptions, we might refer to it as having multiple properties (children are properties of the parent object when they are directly part of its structure and not simply referred to by some other structure that is a property, such as a list).
In describing data structure relationships (such as in a relational database), this would be called a
One-to-Many relationship in terms of the parent object to its children (possibly the closest to a single term, as it is standardized as a compound hyphenation).
"Multiple child objects" are properly described… just as that. Due to the nature of programming, it's far more important to properly describe the relationships as precisely as possible than it is to unnecessarily force brevity in a way that may be confusing or fail to actually be accurate.