What is the term for a shared name (like Henry V, Henry VIII)?

Factual Questions
1

Trying to do a search for a column, but need the name of this term.

Thanks in advance

-NCB-

2

In a monarchical context, that’s called the “regnal name”, with the number being the “regnal number”.

3

Wow. That search word brought up some interesting threads! Thanks, jklann.

What is it called if not a royal family? I have a friend who is a III and he isn’t even close to royal.