If McGee was a Muggleborn who was late to discover his magic, Harry could be the one to introduce him to the wizarding world. Harry would teach McGee about magic, and McGee would in turn show Harry some of the useful Muggle skills he has, like using technology in creative ways to assist in magical tasks.
They could interact as classmates at Hogwarts. McGee might be in the same year as Harry and they study together, help each other with homework, and get into adventures in the castle.
Another way is through a friendly rivalry. Artemis, with his brilliant mind, could compete with Hermione or other intelligent characters in the Harry Potter world in areas like spell - making or magical theory. They could learn from each other's different ways of thinking, with Artemis bringing in scientific thinking and the Hogwarts students sharing their magical knowledge.
Matilda could be invited to Hogwarts as a special guest lecturer by Dumbledore, who has heard of her extraordinary abilities. Harry would be assigned to show her around the castle. As they walk through the corridors, they start discussing the similarities and differences between the magic in Hogwarts and Matilda's own powers. They might also compare the challenges they've faced in their respective worlds, like Harry against Voldemort and Matilda against her cruel parents and the headmistress of her school.
They could interact through a magical portal that connects their two worlds. For instance, Supergirl could be curious about the magic in Harry's world and visit Hogwarts. She might use her superpowers to help in a Quidditch match, like flying really fast to catch the Golden Snitch before any of the players. Harry, on the other hand, could visit Metropolis with Supergirl and be amazed by the tall buildings and advanced technology.
In a fanfiction, Puck could be a sort of guide for Harry in the Forbidden Forest. Harry is often in danger there, and Puck, with his knowledge of nature and magic, could help him navigate through the forest and avoid dangerous creatures. Also, Puck might introduce Harry to some magical creatures that are unique to his world.
They could interact through a magical portal. For example, Mulan could be curious about the strange new world she enters through the portal and Harry could help her understand the wizarding world. He could introduce her to things like wands and spells.
One way for them to interact is if a wizard from the Harry Potter world has a special interest in the magic of Mowgli's jungle. This wizard could send Harry on an expedition there. When Harry arrives, Mowgli sees him as a strange intruder at first. But as Harry shows his good intentions, they start to work together. Mowgli could introduce Harry to the different parts of the jungle, and Harry could share stories of his adventures at Hogwarts. As they spend more time together, they realize that they can help each other. For instance, Mowgli's animal friends could help Harry find a rare magical plant in the jungle, and Harry could use magic to protect the jungle from a group of poachers who are after some of the magical animals.
In a fanfiction, Noctis and his friends could be summoned to Hogwarts by Dumbledore because he senses a great danger approaching that only their combined powers can defeat. Noctis might initially be suspicious of the wizards, but as he gets to know Harry and his friends, he realizes they share the same goal of protecting their respective worlds. They could then train together, with Noctis teaching the wizards some combat techniques and the wizards teaching Noctis about spells.
The characters can interact through a magical accident. For instance, a spell gone wrong in Hogwarts could open a portal to the reaper's world where the Undertaker resides. Harry Potter might be curious about this strange new world and the Undertaker, being an eccentric character, could be intrigued by Harry's magical abilities. They start to exchange knowledge about magic and the reaper's work.
Hermione could use her vast knowledge of magic and research skills to study Orochi. She might try to find a way to communicate with it, thinking that there could be a rational explanation for its existence and actions. Harry, on the other hand, might be more cautious and initially see Orochi as a threat. He would approach it with his Gryffindor courage, but also be ready to defend himself and his friends.
Imagine a scenario where a powerful dark wizard from Harry Potter is trying to gain power from the vampires in Twilight. He kidnaps one of the Cullens and takes them to the wizarding world. This would force the other Cullens and some of the Harry Potter characters, like Harry and his friends, to team up to rescue the kidnapped vampire. During this process, they would have to learn about each other's strengths, weaknesses, and cultures.