No, he’s making it clear that his name is neither Juan nor Carlos, it’s Juan Carlos. He doesn’t continue any of the previous lines of Carloses or of Juanes (either of which was a mess of numerals; note also that at the time of his accesion his father Don Juan was still around and equally ill-considered by pretty much any side you care to name). Felipes on the other hand are numeral-safe, and none of the four Infantas have names that would have been numeral-problematic.
Hey, they are ALL “Saint So-and-So, DEAD Bishop”, “Saint This-and-That, DEAD Confessor”. One of the major (or mild, if you are a true believer) inconveniences of gaining that accolade.
I, too, find merit in the explanation that the 1970s compound-name adopters (the first Pope John Paul and King Juan Carlos) would have deliberately chosen to include the ordinal so as to make it clear they were not to be part of the sequence of prior Johns, Pauls, Juanes or Carloses. The Spanish King in particular as a way of emphasizing a new beginning. (More currently, Willem-Alexander of the Netherlands reigns with a compound *hyphenated *name and follows the traditional style of not using an ordinal)
I don’t know in general, but there three St Augustines (or should that really be Saints Augustine), known respectively as
[ul]
[li]St Augustine of Hippo (the city where he was bishop) (AKA the great, or greater)[/li][li]St Augustine of Canterbury (where he was bishop) (AKA the less, or lesser)[/li][li] and St Augustine Webster. He is of much less historical importance than either of the others, never got to a Bishop, and became a saint after the Reformation (indeed, was killed during it, by Henry VIII’s guys), which means he is only a saint for Catholics, and not for Protestants at all.* Unlike the other two, however, he did have a surname, so he gets differentiated by that (when he gets mentioned at all).[/li][/ul]
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*I seem to remember reading somewhere that Francis of Assisi is the last, i.e., most recent, saint to be recognized as such by Protestants. Is that right? Surely there must have been other saints canonized in the ~300 years between Francis and Martin Luther? What did they do wrong?
This is the form I would use. Just as there have been VI Kings George so far, and XXIII Popes John, and 2 Presidents Roosevelt.
ETA: And, I think, VII Queens Cleopatra. 
Well, he did author the Holy Dictionary.