Consider the name John J Doe Jr. If John is the first name, J is the middle initial, & Doe is the last name, what is Jr.? I checked with Webster years ago and they could not find a specific name.
It is a generational title/suffix:
Generational suffixes are used to distinguish persons who share the same name within a family. A generational suffix can be used informally (for disambiguation purposes, or as nicknames) and is often incorporated in legal documents.