In episode 3, many argue for Aegon to be the next ruler of Westeros (he is the firstborn son of King Viserys), while others insist the throne is still Rhaenyras (he is the actual firstborn son of King Viserys), and he wants a white stag to greet him around the Kingswood.
But when Viserys (Paddy Considine) is called to a stag, the result isn't what he expected. It's, as one of the helpers notes, still a big lad, but the animal isn't white. This moment, which is staged, is underwhelming, and lacks the clear symbolism he so clearly seeks, provides no clarity about who the gods are willing to show their favor.
Rhaenyra (Milly Alcock) is the one who notices the gigantic white stag on a cliff in the morning. By design, the scene feels a bit mystical, tapping into the kind of magic Game of Thrones used to wield in the future.
The world of A Song of Ice and Fire is unpredictable and random. No religion had a clear grasp on magic, but none of them had a clear grasp on it. Changed faces, reanimation, and ghost demon babies all existed with the same predictability as lightning or a vision from looking into a fire.
House of the Dragon is based on Fire & Blood, which provides a fairly brief account of the events as told by three people who witnessed (or witnessed) them. While George R.R. Martin brought whole characters back to life in the A Song of Ice and Fire books that the show left dead, Fire & Blood reads more like a textbook, eliminating the more otherworldly aspects of Martins' world.
The stag hunt in episode 3, like the Aegon the Conquerors' dream from episode 1, is a step back towards the more supernatural world of Thrones. This time the story may be one we know the end of, but the signs along the way are more ambiguously otherworldly. The Iron Throne seems to reject certain occupants, and the white stag appears to those who may be worthy.