Śākya Gaṇarājya translates to English as Shakya Republic and that is actually correct. Kapilavastu was the capital of the Republic/Oligarchy of the Shakya clan. Buddha's father was the elected chief at the time of his birth. This is why one of his epithets is Shakyamuni ("muni" means sage so sage of the Shakyas.)

Strictly speaking Buddha was not a royal price but scion of an aristocratic family. As a comparison think of someone born in a patrician family of Rome when his father is a Consul of Rome. The Wikipedia article has good details about the Shakyas: https://en.wikipedia.org/wiki/Shakya