California, it's an American custom! I cannot speak for the other European countries, but we are not familiar with this numbering practise in Denmark. You could add senior or junior, though, if they were father and son!
Viv
In English, "II" is normally used for a boy named after his uncle or grandfather. If he's named for his father, he'd be called, "Jr."
If the son, father, and grandfather all had the same names, the son would be "III." It goes on IV, V, etc.
The tradition here is that men use a suffix only if all their names are identical to their father's (or uncle or grandfather). That's why George W. Bush isn't a "junior"-- his father is George H. W. Bush. So, little Henrik wouldn't be "junior" or a "II."
(In proper form, "Sr." is used only by the widow of the father, never by the man himself. For example, after her husband died, Jackie Kennedy was "Mrs. John F. Kennedy, Sr.")
Anyway, that's probably more than you wanted to know!
Short version: The media here would probably call him Henrik II or Henrik Jr. regardless of the proper form!