His personality might be more independent and self - reliant. Without the constant presence of the village and its social structures, he has to trust his own instincts more. So he becomes less likely to seek approval from others.
Naruto's personality would likely be more introverted. Instead of seeking constant attention and approval from the village, he might find solace in the company of the Kitsune and the natural world around him. His sense of justice could also be more complex. He might not simply see things in terms of good and bad as he did in the original story. For example, he could understand that some enemies might have reasons for their actions due to his Kitsune - given wisdom. His relationships with others would also be different, as he would approach people with a more cautious and observant eye, having learned from the Kitsune's solitary yet perceptive nature.
Naruto's personality would develop in a more complex way. With Kurenai as a mother figure, he'd be more empathetic and in tune with others' emotions. Anko would add a bit of rebelliousness to his character. He'd still be determined like in the original story, but his way of achieving his goals would be more refined. He might be more secretive too, as he'd learn from Anko that not everything needs to be out in the open all the time.
He would become more reliant on his friends and allies. Since he doesn't have chakra to fall back on, he has to trust others more and learn to work in a team in a different way.
Naruto's confidence would likely be different. Instead of the somewhat brash confidence that came from constantly proving himself in the original story, he would have a more serene and assured confidence, based on Hagoromo's teachings. His understanding of the world would be more profound from the start. He wouldn't view enemies as simply 'bad guys' but rather as products of a flawed system. His sense of justice would be more complex and far - reaching. He would strive for a more global peace rather than just the protection of his village. Also, his relationship with Sasuke would be less about rivalry and more about shared understanding. They would both be on a path towards a greater good, and Naruto would be more able to communicate his ideas to Sasuke, preventing some of the rifts that occurred in the original story.
Naruto's personality could become more confident and stable. With a mother figure in his life, he would have someone to turn to for emotional support. He might not be as impulsive as he was before because Kushina could give him advice based on her own experiences. For example, when dealing with conflicts in the village or with other ninja, he would think twice instead of jumping into action right away.
She would likely be more influenced by Japanese values such as respect for elders and harmony. So, instead of being as headstrong as the original Harry at times, she might be more diplomatic in her approach to problems. For instance, when facing a bully in the Japanese wizarding school, she might first try to resolve the issue through peaceful means rather than immediately retaliating.
Harry would be more cautious. Moody's paranoia would rub off on him. He'd be constantly on guard, not as quick to trust people like in the original series. For example, when he meets new classmates, he might be sizing them up for potential threats instead of being friendly right away.
Naruto might develop a more cynical view of the world at first. Since Obito is involved in his 'founding', there could be darker elements in his upbringing. He may start off being more distrustful of others. However, as he discovers the truth about Obito's actions, he could use that knowledge to grow into a more complex and empathetic character, different from the naive but positive Naruto in the original story.
Since McGonagall has a strong sense of justice, Harry would probably have an even more acute sense of right and wrong. He would grow up with her values of fairness and loyalty deeply ingrained. This could lead to him being more proactive in standing up for others who are being wronged, not just because of his own experiences, but because of the values he was taught by McGonagall.
His personality would likely be more assertive and less naive. Anko's boldness and Kurenai's intelligence would rub off on him. He'd be quicker to make decisions and not as easily swayed.