He wasn't born into royalty, but he was adopted.

Long story short: When he was young (about 90, barely into his teens compared to humans) he visited a planet that wasn't very technologically advanced (like Earth during the 1900's) and a rival kingdom was poisoning the capitol's water supply. Drax used the medical equipment onboard his freighter to make an antidote, saving the royal family and city. The King was so grateful that he adopted Drax since he was so young (they have dragons on that planet so the King knew about the age comparison) and made him a Prince. Drax had no claim to the throne because the King had a son already.

When Drax and his family visited the planet later (100 years or so), the two kingdoms were in a full blown war. The current King (Drax's step brother who was almost 110 years old) abdicated, and so did his son so Drax could be crowned as King.
Because Drax used one of his destroyers (https://www.furaffinity.net/full/5108137) to visit, it was used to defend the small kingdom. Drax accepted the title and throne on one condition: That his step-nephew and his descendants reign in his absence, since Drax would be off planet and lead his mercenaries.

Bonus info: Drax and Kietsu hooked up during this campaign, and got married later with Krrreeshna's blessing