"Fatherhood" is German, "paternity" is Latin, in origin. English sometimes has two or even three words for the same thing, from Germanic, Latin directly, and Latin through medieval French. Sometimes they have differences in meaning, sometimes they don't. Here, as is often the case with the Latin or French term, it is being used in a legal context while the original Germanic term is used in a more "popular" or personal sense.
"Royal" is the legal term for "belonging to or pertaining to the King/monarch"—all those British government institutions described as "Royal ____." "Kingly" and "regal" have to do with appearance and attitude, and one can describe someone who is not a king (or an animal, for that matter) as "kingly" or "regal." In modern German, their version of "kingly" is used where English uses "royal."
There is a similar situation to paternity/fatherhood in English with fraternity/brotherhood. "Brotherhood" means the same as "fraternité" in the Revolutionary slogan, although English also uses "fraternity" for that meaning. But in English "fraternity" is the only one of the pair used for the formal friendship organization at American universities; the earliest ones were formed when Latin was still widely studied in such institutions. American universities have "fraternities" and "sororities," but not "brotherhoods" or "sisterhoods."