When speaking of linguistics, and onomastics is a part of linguistics, "legitimate" has to be understood as "conforming to recognized principles or accepted rules and standards", being this principles, rules and standards purely linguistic ones and not civil law.
That is, a name can be legal (accepted by the civil law to be registered) and not be legitimate because it doesn't conform to the language rules, spelling, for instance. In the same way, a name can be illegal even being completely legit.
In the USA all the names can be legal, but not all of them are legitimate. The legitimacy or not doesn't depend on civil tribunals or laws, but on linguistic rules.