His communication style could be different too. He might be more direct and simple in his speech, like how animals communicate basic needs and emotions. He wouldn't be as interested in the complex social hierarchies and politics among demigods. Instead, he'd focus on the more straightforward relationships based on mutual respect and survival, much like the relationships in the animal kingdom.
He'd probably be more brooding. Growing up with Hades, he'd be exposed to the darker side of the mythological world constantly. So, instead of the somewhat upbeat Percy we know, he might be more withdrawn and moody.
Percy's personality would be more in tune with nature. Since Artemis has a strong connection to nature, Percy would grow up respecting and understanding it more deeply. He would probably be more solitary as well, as the hunters often keep to themselves. He might not be as quick to trust others as he used to be, having been brought up in a more exclusive group. Also, he would have a greater sense of loyalty towards Artemis and the hunt, which could make him seem a bit cold to outsiders.
Percy could be more in tune with his emotions. Dionysus is associated with passion and ecstasy, so Percy might be more openly emotional, not suppressing his feelings as much as he did in the original stories. He could also be more creative, as Dionysus has an element of creativity in his domain. For example, he might be better at coming up with unique solutions to problems using his more artistic side.
He'd be more in tune with the sea. He might be calmer, like the ebb and flow of the ocean. His reactions to things would be more measured as he's used to the slow, powerful nature of the sea. For example, instead of getting angry quickly, he'd wait and think like the sea does before a big wave hits.
He'd probably be more sophisticated. The Delacour family is known for their elegance and grace. Harry might have a more refined way of speaking and acting compared to his canon self. For instance, he would be more likely to use formal French wizarding greetings.
His personality would likely be more reserved and calculating. Growing up with the Malfoys, he'd be taught to keep his emotions in check and think before acting. He might be more aloof towards others outside of the pure - blood circle at first.
He would probably be more in tune with nature. Since animals operate on instinct a lot, he might have a stronger sense of intuition. For example, if he was raised by deer, he'd be more skittish and aware of his surroundings, always on the lookout for danger like they are.
Percy's personality would likely become more hardened. Growing up with assassins, he'd see a lot of violence and death. This would make him less trusting and more cynical. He might not be as quick to make friends as he was in the original story. However, his loyalty to those he does trust would be even stronger, like the loyalty assassins have to their comrades. Also, he'd be more disciplined, following the assassin code he was raised with.
Percy's personality would probably be more solitary at first. Growing up with wolves, he's used to relying on himself and his pack. So when he enters the world of demigods, he might have trouble trusting others immediately. He'd be more in tune with his instincts, which could make him seem unpredictable to those who don't understand his background. For example, he might follow his nose or gut feeling more often than following the advice of others.
He'd likely be more reserved. Artemis values solitude and purity, so Percy might be less of a joker. He'd be more serious about his duties and tasks.