Through shared experiences. For instance, they could participate in a special event in the game world that forces them to trust and depend on each other. This would be a great start to developing their relationship.
Maybe Elizabeth discovers a prophecy about Minato and she becomes determined to protect him at all costs. This could lead to a lot of action - packed scenes as well as some really tender moments between them. For example, when she has to shield him from a powerful enemy attack.
If we consider the general 'Elizabeth story', Elizabeth is likely the central figure. Her significance could be that she drives the plot forward. For example, if it's a story of her journey, her decisions and actions would be what the story is built around.
In 'Persona 4 Visual Novel', the gay persona might represent the diversity of the game's character creation and the exploration of different sexual orientations within the game's fictional world. It could add more depth to the relationships and storylines.
Perhaps the fanfiction focuses on the cultural clash between the two groups. The Persona 4 characters are from a small town, while the Persona 5 crew is from the big city. Their different ways of handling things and their unique Personas could lead to some interesting interactions. For example, Yu and Joker might have different leadership styles that they have to reconcile in order to succeed against the shadows.
Maybe the Velvet Room attendants from both games somehow get their worlds mixed up. Igor from Persona 5 and Margaret from Persona 4 have to work together to set things right. This leads to the characters from both games interacting. The Phantom Thieves might learn some new combat techniques from the Investigation Team and vice versa. They could then use these new skills to face a common threat that emerges from the merging of their two realities.
Maybe there's a strange distortion that merges parts of Inaba and Tokyo. The characters from both Persona 4 and Persona 5 find themselves in this strange new place. They start to notice that the Shadows in this merged world are more powerful and have new abilities. The Investigation Team and the Phantom Thieves must figure out how to adapt their Personas and strategies to fight these new threats while also dealing with the cultural differences between their two groups.