In such a fanfiction, Harry being born a phoenix could mean he has a strong connection to magic that is even more primal than in the original story. He could have an innate understanding of magic related to life and death, given the phoenix's symbolism in the wizarding world. For example, he might be able to sense when someone is near death or in great danger, and use his phoenix - born powers to save them. Also, his presence could bring a sense of renewal to Hogwarts, as if he's a symbol of hope and rebirth for the school.
A 'Harry Potter is born a phoenix' fanfiction would likely change a lot of the dynamics. Since phoenixes are known for their loyalty, Harry might have an even stronger sense of loyalty to his friends from the very start. His physical form could also be different. He might be more resilient to physical harm, and his body temperature could be a bit higher than normal, giving him a sort of 'fiery' nature. And when he gets angry or determined, there could be a faint glow around him, similar to the flames of a phoenix.
Well, it could be really interesting. Maybe Harry would have some phoenix - like abilities from birth. He might be able to heal himself quickly like a phoenix does when it rises from the ashes. And his eyes could have a special, fiery glint.