That analogy is broken; it's traditional to use Roman numerals to indicate that people hold the nth instance of the same name in a line of patrilineal descent (or, even more traditionally, a line of rulership). Roger White's son could be Roger White II; Roger White IV's son is definitely Roger White V.
But two Roger Whites from separate families won't use numbers at all; both of them are just Roger White. The AKC is numbering unrelated dogs.
No descent is implied when numbers are used for monarchs and popes. John XXIII was the 23rd pope named John (20th century), a long time after John XXII in the 14th century. Queen Elizabeth II of England (current) is not a direct descendant from Queen Elizabeth I (16th century).
Descent is implied, just not genealogical descent. John XXIII is the 23rd Pope John in the same line of papacy. Pope Benedict XIII (17th century) is the 13th Pope Benedict in that line, but Pope Benedict XIII (14th-15th century) was the 13th Pope Benedict in the line of Popes in Avignon. Henry IV of England was the fourth King Henry of England; Henry IV of France was the fourth King Henry of France; like the Popes, they are two different people tracked in two different lines of descent.
But two Roger Whites from separate families won't use numbers at all; both of them are just Roger White. The AKC is numbering unrelated dogs.
And we appear to use arabic numerals for that use case: see https://www.espn.com/golf/story/_/id/26880947/lee6-wins-us-w...