The race we know as the Hive didn’t exist until after the sisters Aurash, Xi Ro, and Sathona accepted the bargain with the worm gods. This bargain involved taking the larvae of the worm gods into their bodies, allowing them to form a symbiotic relationship with the larvae. The sisters, now known as Auryx, Xivu Arath, and Savathûn, then took the larvae to the rest of their species. Those who accepted the larvae became the Hive, while those who refused were killed. In return for accepting the bargain, the Hive were given immortality and great power.
Prior to this bargain, Aurash, her sisters, and the rest of their race were a species known as the proto-hive. The proto-hive had a lifespan of only ten years, except for those who were “mothers”, who could live for much longer. Oryx’s father, the Osmium King, was one of the proto-Hive, not the Hive. There is no mention of what the Osmium King looked like, although the Helium Drinkers, who were also of the proto-hive species, are described as having “two legs, two arms, and three eyes”. It’s likely, although we can’t be sure, that the Osmium King looked like these Helium Drinkers and didn’t have any wings.
It’s not clear if Oryx gained wings when he took on the king morph, or whether he grew them at a later time. In XXXV: This Love is War, Xivu Arath says, “Although he was born at the bottom of the universe, and taught to burrow, Oryx has grown wings.” XXXVIII: The partition of death says, “One day Oryx decided to grow new wings.” This suggests that Oryx may have changed his appearance more than once after taking the king morph, so perhaps the wings were a later addition. There is further possible evidence that the wings may not have been there when Oryx first took the king morph in XLIII: End of Failed Timeline. In this card, Quria, a Vex mind, tries to simulate Oryx but is only able to simulate Aurash, the proto-hive form of Oryx before he accepted the bargain with the worm gods and took on the king morph. This simulated Aurash was “Oryx minus the symbiote organism, minus the wings and morphs, minus the weapon, minus the power.” The wings being mentioned separately from the morphs may suggest that Oryx did not gain the wings when he took on the king morph but instead grew them later.