In episode 3, there isnt a lot of agreement on who should be the next ruler of Westeros. Some people want Aegon to be the next in line (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)
When Viserys (Paddy Considine) is called to a stag, it isn't what he imagined. It's, as one of the helpers notes, still a big lad, but the animal isn't white. This moment is staged, underwhelming, and lacking the clear symbolism he so clearly seeks, providing no clarity about who the gods will show their favor to.
Rhaenyra (Milly Alcock) is the one who spots the enormous white stag at a cliff in the morning. She stares it down, prevents Ser Criston (Fabien Frankel) from killing it, and lets it go away. The whole scene feels a bit mystical, tapping into the type of magic Game of Thrones used to employ to keep its high fantasy characters guessing about the future.
The world of A Song of Ice and Fire is unpredictable and randomly spread throughout the world. No religion had a clear grasp on magic, but none of them had a clear grasp on it. There were changing faces, reanimation, and ghost demon babies, all of whom existed with the same degree of predictability as lightning or a vision from looking into a fire.
House of the Dragon is a biography of three people who witnessed (or witnessed) the events. 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, as opposed to Aegon the Conquerors' fantasy from episode 1, is a step back toward a more alternate reality like 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.